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Similar Question 1
<p>Consider the following data set.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|c|c|c|c|c|} \hline \mathrm{x} & -3 & -1 & 1 & 3 & 5 & 7 \\ \hline \mathrm{y} & 5 & -5 & -3 & 5 & 6 & 45 \\ \hline \end{array} </code></p><p>Estimate the instantaneous rate of change at the point corresponding to each x-value.</p> <ul> <li>i) <code class='latex inline'>x = -1</code></li> <li>ii) <code class='latex inline'>x =1</code></li> <li>iii) <code class='latex inline'>x = 3</code></li> <li>iv) <code class='latex inline'>x = 5</code></li> </ul>
Similar Question 2
<p>For each of the functions in question 5, estimate the instantaneous rate of change at <code class='latex inline'>x = 3</code>.</p><p><code class='latex inline'>f(x)=3x+1</code></p>
Similar Question 3
<p>Table of values are from the function <code class='latex inline'>f(x) = 5x^2 - 7</code> below.</p><img src="/qimages/119" /><p>Based on the trend in the average rates of change, what is an estimate of the instantaneous rate of change when <code class='latex inline'>x = 2</code>?</p>
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Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
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<p>Estimate the instantaneous rate of change of <code class='latex inline'>f(x)=\displaystyle{\frac{x}{x-4}}</code> at the point <code class='latex inline'>(2,-1)</code>.</p>
<p>On Earth, the height, <code class='latex inline'>h</code>, in metres, of a free-falling object after t seconds ca be modelled by the function <code class='latex inline'>h(t) = -4.9t^2 + k</code>, while on Venus, the height can be modelled by <code class='latex inline'>h(t) = -4.45t^2 + k</code>, where <code class='latex inline'>t \geq 0</code> and <code class='latex inline'>k</code> is the height, in metres, from which the object is dropped. Suppose a rock is dropped from a height of 60 m on each planet.</p> <ul> <li>Estimate the instantaneous rate of change of the height of the rock 3 s after it is dropped.<br></li> </ul>
<p>The movement of a certain glacier can be modelled by <code class='latex inline'>d(t) = 0.01t^2 + 0.5t</code>, where <code class='latex inline'>d</code> is the distance, in metres, that a stake on the glacier has moved, relative to a fixed position, <code class='latex inline'>t</code> days after the first measurement was made. Estimate the rate at which the glacier is moving after <code class='latex inline'>20</code> days.</p>
<p>The demand function for snack cakes at a large bakery is given by the function <code class='latex inline'>p(x)=\displaystyle{\frac{15}{2x^2+11x+5}}</code>. The x-units are given in thousands of cakes, and the price per snack cake, <code class='latex inline'>p(x)</code>, is in dollars.</p><p>Find the revenue function for the cakes.</p>
<p>Graph the function <code class='latex inline'>f(x)=x^3-2x^2+x</code> by finding its zeros. </p><p>Use the graph to estimate where the instantaneous rate of change is positive, negative, and zero.</p>
<p>Determine the slope of the line that is tangent to the graph of each function at the given point. Then determine the value of <code class='latex inline'>x</code> at which there is no tangent line.</p><p><code class='latex inline'>f(x)=\displaystyle{\frac{-5x}{2x+3}}</code>, where <code class='latex inline'>x=2</code></p>
<p>The height of a football that has been kicked can be modelled by the function</p><p><code class='latex inline'>\displaystyle h(t) = -5t^2 + 20t + 1 </code> , where h(t) is the height in metres and t is the time in seconds.</p><p>a) What is the average rate of change in height on the interval <code class='latex inline'>0\leq t \leq 2</code> and on the interval <code class='latex inline'>2 \leq t \leq 4</code>?</p><p>b) Use the information given in part a) to find the time for which the instantaneous rate of change in height is 0 m/s. Verify your response.</p>
<p>A population of raccoons moves into a wooded area. At <code class='latex inline'>t</code> months, the number of raccoons, <code class='latex inline'>P(t)</code>, can be modeled using the equation <code class='latex inline'>P(t) = 100 + 30t + 4t^2</code>.</p><p>What is the population of raccoons at <code class='latex inline'>2.5</code> months?</p>
<p>What are the slopes of these secant lines?What is an estimate of the slope of the tangent line when <code class='latex inline'>x = 2</code>?</p><p><code class='latex inline'> f(x) = 3x^2 - 5x + 1 </code></p>
<p>A soccer ball is kicked into the air. The following table of values shows the height of the ball above the ground at various times during its flight.</p><img src="/qimages/120" /><p><strong>(a)</strong> What is an estimate of the instantaneous rate of change in the height of the ball at exactly <code class='latex inline'>t = 2.0</code> s using the preceding and following interval method?</p><p><strong>(b)</strong> What is an estimate of the instantaneous rate of change in the height of the ball at exactly <code class='latex inline'>t = 2.0</code> s using the centered interval method?</p>
<p>What do all the slopes in each set of functions have in common?</p> <ul> <li><code class='latex inline'>f(x) = -x^2 + 6x - 4</code> when <code class='latex inline'>x= 3</code></li> <li><code class='latex inline'>g(x) = \sin x</code> when <code class='latex inline'>x= 90^o</code></li> <li><code class='latex inline'>g(x) = x^2 + 4x + 11</code> when <code class='latex inline'>x= -2</code></li> <li><code class='latex inline'>f(x) = 5</code> when <code class='latex inline'>x= 1</code></li> </ul>
<p>An oil tank is being drained. The volume, <code class='latex inline'>V</code>, in litres, of oil remaining in the tank after time, <code class='latex inline'>t</code>, in minutes, is represented by the function <code class='latex inline'>V(t) = 60(25 - t)^2</code>, <code class='latex inline'>0\leq t \leq 25</code>.</p> <ul> <li>Determine the average rate of change of the volume during the first 10 min, and then during the last 10 min. Compare these values, giving reasons for any similarities and differences.</li> </ul>
<p>Consider the following data set.</p><p><code class='latex inline'>\displaystyle \begin{array}{|c|c|c|c|c|c|c|} \hline \mathrm{x} & -3 & -1 & 1 & 3 & 5 & 7 \\ \hline \mathrm{y} & 5 & -5 & -3 & 5 & 6 & 45 \\ \hline \end{array} </code></p><p>Estimate the instantaneous rate of change at the point corresponding to each x-value.</p> <ul> <li>i) <code class='latex inline'>x = -1</code></li> <li>ii) <code class='latex inline'>x =1</code></li> <li>iii) <code class='latex inline'>x = 3</code></li> <li>iv) <code class='latex inline'>x = 5</code></li> </ul>
<p>Use an algebraic strategy to verify that the point given for the function is either a maximum or minimum.</p><img src="/qimages/157680" /><p><code class='latex inline'>\displaystyle f(x)=-x^{3}+12 x-1 ;(2,15) </code></p>
<p>The demand function for snack cakes at a large bakery is given by the function <code class='latex inline'>p(x)=\displaystyle{\frac{15}{2x^2+11x+5}}</code>. The x-units are given in thousands of cakes, and the price per snack cake, <code class='latex inline'>p(x)</code>, is in dollars.</p><p>Estimate the marginal revenue for <code class='latex inline'>x=0.75</code>. What is the marginal revenue for <code class='latex inline'>x=2.00?</code></p>
<p>Use an algebraic strategy to verify that the point given for the function is either a maximum or minimum.</p><p><code class='latex inline'>\displaystyle f(x)=x^{2}-9 x ;(4.5,-20.25) </code></p>
<p>Estimate the instantaneous rate of change of the function in question 1 at <code class='latex inline'>x=2</code> by determining the slope of a secant line from the point where <code class='latex inline'>x=2</code> to the point where <code class='latex inline'>x=201</code>. </p><p>Compare your answer with your answer for question 1, part b).</p>
<p>Determine the instantaneous rate of change at <code class='latex inline'>x= 2</code>.</p><p><code class='latex inline'>\displaystyle y =x^2 +3x</code></p>
<p>A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.</p><p><code class='latex inline'>h(x) = -2x^2 + 68x + 75; (17, 653)</code></p>
<img src="/qimages/11577" /><p>Estimate the slope of the tangent line in the graph of this function.</p>
<p>A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.</p><p><code class='latex inline'>j(x) = \sin(-2x); (45^o, -1)</code></p>
<p>Estimate the instantaneous rate of change at the tangent point indicated on each graph.</p><img src="/qimages/7143" />
<p>For each of the functions in question 5, estimate the instantaneous rate of change at <code class='latex inline'>x = 3</code>.</p><p> <code class='latex inline'>t(x)=3x^2-4x+1</code></p>
<p>How would the graph of this function change in question <strong>A</strong> if the ladybug landed on a spot 1 m form the tip of the blade? What effect would this have on the rate of change of the height? Support your answer.</p> <ul> <li><strong>A</strong> The blades of a particular windmill sweep in a circle 10 m in diameter. Under the current wind conditions, the blades make one rotation every 20s. A ladybug lands on the tip of one of the blades when it is hat th bottom of tis rotation, at which point the ladybug is 2 m off the ground. It remains on the blade for exactly two revolutions, and then flies away.</li> </ul>
<p>Consider the function <code class='latex inline'>f(x)=3x^2-4x-1</code>.</p><p>Estimate the slope of the tangent line at <code class='latex inline'>x=1</code>.</p><p>Find the y-coordinate of the point of tangency.</p><p>Use the coordinates of the point of tangency and the slope to find the equation of the tangent line at <code class='latex inline'>x=1</code>.</p>
<img src="/qimages/11577" /><p>Estimate the slope of the tangent line in the graph of this function.</p><p>What does the slope of the tangent represent?</p>
<p>Identify whether each situation represents an average rate of change or an instantaneous rate of change. Explain your choice.</p> <ul> <li>When the area of a circular ripple on the surface of a pond is 4 cm, the circumference of the ripple is increasing at 21.6 cm/s.</li> </ul>
<p>Determine the instantaneous rate of change at <code class='latex inline'>x= 2</code>.</p><p><code class='latex inline'>\displaystyle y =7x^2 -x^4</code></p>
<p>How would the graph of the height of the ladybug in question <strong>A</strong> change if the wind speed increased? How would this graph change if the wind speed decreased? What effect would these change have on the rate of change of the height of the ladybug? Support your answer. </p> <ul> <li><strong>A</strong> The blades of a particular windmill sweep in a circle 10 m in diameter. Under the current wind conditions, the blades make one rotation every 20s. A ladybug lands on the tip of one of the blades when it is hat th bottom of tis rotation, at which point the ladybug is 2 m off the ground. It remains on the blade for exactly two revolutions, and then flies away.</li> </ul>
<p>A population of raccoons moves into a wooded area. At <code class='latex inline'>t</code> months, the number of raccoons, <code class='latex inline'>P(t)</code>, can be modelled using the equation <code class='latex inline'>P(t) = 100 + 30t + 4t^2</code>.</p><p>What is the average rate of change in the raccoon population over the interval from <code class='latex inline'>0</code> months to <code class='latex inline'>2.5</code> months?</p>
<p>Consider the function <code class='latex inline'>f(x)=x^3-4x^2+4x</code>.</p><p>Estimate the instantaneous rate of change in <code class='latex inline'>f(x)</code> at <code class='latex inline'>x = 2</code>.</p>
<p>Explain how instantaneous rates of change could be used to locate the local maxima and local minima for a polynomial function.</p>
<p>The height, <code class='latex inline'>h</code>, in metres of a toy rocket above the ground can be modelled by the function <code class='latex inline'>h(t)=-5t^2+50t</code>, where <code class='latex inline'>t</code> represents time in seconds.</p><p>Use an average speed to approximate the instantaneous speed at <code class='latex inline'>t=4</code>.</p><p>Use an average speed to approximate the instantaneous speed at <code class='latex inline'>t=10</code>.</p><p>What is the average speed over the interval from <code class='latex inline'>t=0</code> to <code class='latex inline'>t=10</code>?</p>
<p>A firework is shot into the air such that its height, h, in metres, after 1 seconds can be modelled by the function <code class='latex inline'>h(t) = -4.9t^2 + 27t + 2</code></p><p> Copy and complete the table.</p><img src="/qimages/307" /> <ul> <li> Use the table to estimate the velocity of the firework after 3 s.</li> </ul>
<p>Use an algebraic strategy to verify that the point given for the function is either a maximum or minimum.</p><p><code class='latex inline'>f(x) = x^2 - 4x + 5</code> at <code class='latex inline'>(2, 1)</code></p>
<p>Determine the slope of the line that is tangent to the graph of each function at the given point. Then determine the value of <code class='latex inline'>x</code> at which there is no tangent line.</p><p><code class='latex inline'>f(x)=\displaystyle{\frac{5}{x-6}}</code>, where <code class='latex inline'>x=4</code></p>
<p>For each of the functions in question 5, estimate the instantaneous rate of change at <code class='latex inline'>x = 3</code>.</p><p><code class='latex inline'>h(x)=2^x</code></p>
<p>For each of the functions in question 5, estimate the instantaneous rate of change at <code class='latex inline'>x = 3</code>.</p><p><code class='latex inline'>v(x)=9</code></p>
<p>Use an algebraic strategy to verify that the point given for the function is either a maximum or minimum.</p><p><code class='latex inline'>\displaystyle f(x)=3 \cos (x) ;\left(0^{\circ}, 3\right) </code></p>
<p>Estimate the instantaneous rate of change at <code class='latex inline'>x = -3</code> as follows:</p><p><strong>(a)</strong> Substitute <code class='latex inline'>h = 0.1, h = 0.01</code>, and <code class='latex inline'>h = 0.001</code> into the expression and evaluate.</p><p><strong>(b)</strong> Simplify the expression, and then substitute <code class='latex inline'>h =0.1, h=0.01</code>, and <code class='latex inline'>h =0.001</code> and evaluate.</p>
<p>For the function <code class='latex inline'>f(x) = 6x^2 - 4</code>, what is an estimate of the instantaneous rate of change for the given value of <code class='latex inline'>x</code>?</p><p><code class='latex inline'>x = 4</code></p>
<p>Identify whether each situation represents an average rate of change or an instantaneous rate of change. Explain your choice.</p> <ul> <li>Nike travels 550 km in 5 h.</li> </ul>
<p>Use the slopes to estimate the slope of the tangent line when <code class='latex inline'>x = 2</code>.</p><p><code class='latex inline'> f(x) = 2x - 7 </code></p>
<p>A submarine is descending from the surface. What is the best estimate of its instantaneous change in depth at <code class='latex inline'>\displaystyle t=3 \mathrm{~s} </code> ? </p><img src="/qimages/157698" /><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline Time, \boldsymbol{t}(\mathbf{s}) & Depth (\mathbf{m}) \\ \hline 3 & 27 \\ \hline 3.001 & 27.015002 \\ \hline 3.01 & 27.1502 \\ \hline 3.1 & 28.52 \\ \hline 4 & 44 \\ \hline \end{array} </code></p><p>a) <code class='latex inline'>\displaystyle 15.2 \mathrm{~m} / \mathrm{s} </code></p><p>b) <code class='latex inline'>\displaystyle 15.002 \mathrm{~m} / \mathrm{s} </code></p><p>c) <code class='latex inline'>\displaystyle 15 \mathrm{~m} / \mathrm{s} </code></p><p>d) <code class='latex inline'>\displaystyle 17 \mathrm{~m} / \mathrm{s} </code></p>
<p>For the function <code class='latex inline'>f(x) = 6x^2 - 4</code>, what is an estimate of the instantaneous rate of change for the given value of <code class='latex inline'>x</code>?</p><p><code class='latex inline'>x = 0</code></p>
<p>On Earth, the height, <code class='latex inline'>h</code>, in metres, of a free-falling object after t seconds ca be modelled by the function <code class='latex inline'>h(t) = -4.9t^2 + k</code>, while on Venus, the height can be modelled by <code class='latex inline'>h(t) = -4.45t^2 + k</code>, where <code class='latex inline'>t \geq 0</code> and <code class='latex inline'>k</code> is the height, in metres, from which the object is dropped. Suppose a rock is dropped from a height of 60 m on each planet.</p><p>Determine the average rate of change of the heigh of the rock in the first 3 s after it is dropped.</p>
<p>Determine the slope of the line that is tangent to the graph of each function at the given point. Then determine the value of <code class='latex inline'>x</code> at which there is no tangent line.</p><p><code class='latex inline'>f(x)=\displaystyle{\frac{x-6}{x+5}}</code>, where <code class='latex inline'>x=-7</code></p>
<p>For each of the functions in question 5, estimate the instantaneous rate of change at <code class='latex inline'>x = 3</code>.</p><p><code class='latex inline'>f(x)=3x+1</code></p>
<p>A soccer ball is kicked into the air such that its height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds can be modelled by the function <code class='latex inline'>h(t) = -4.9t^2 +12t + 0.5.</code></p><p>Estimate the instantaneous rate of change of the height of the ball after 1s.</p>
<p>In investments value, <code class='latex inline'>V( t)</code>, is modelled by the function <code class='latex inline'>V(t) = 2500(1.15)^t</code>, where <code class='latex inline'>t</code> is the number of years after funds are invested.</p><p>Find the instantaneous rate of change in the value of the investment at <code class='latex inline'>t = 4</code>.</p>
<p>Estimate the instantaneous rate of change at the tangent point indicated on each graph.</p><img src="/qimages/7142" />
<p>When electricity flows through a certain kind of light bulb, the voltage applied to the bulb, in volts, and the current flowing through it, in amperes, are as shown in the graph. The instantaneous rate of change of voltage with respect to current the light bulb.</p> <ul> <li>Use the graph to determine the resistance of the light bulb at a voltage of 60 V. </li> </ul> <img src="/qimages/992" />
<p>Select a strategy to estimate the instantaneous rate of change of each function at the given point.</p><p><code class='latex inline'>y=\displaystyle{\frac{17x+3}{x^2+6}}</code>, where <code class='latex inline'>x=-5</code></p>
<p>A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.</p><p><code class='latex inline'>m(x) = \frac{1}{20}(x^3 + 2x^2 -15x); (-3, \frac{9}{5})</code></p>
<p>Select a strategy to estimate the instantaneous rate of change of each function at the given point.</p><p><code class='latex inline'>y=\displaystyle{\frac{x+3}{x-2}}</code>, where <code class='latex inline'>x=4</code></p>
<p>The top of a flagpole sways back and forth in high winds. The function <code class='latex inline'>f(t) = 8\sin(180^ot)</code> represents the displacement, in centimetres, that the flagpole aways from vertical, where <code class='latex inline'>t</code> is the time in seconds.</p><p>The flagpole is vertical when <code class='latex inline'>f(t) =0</code>. It is <code class='latex inline'>8</code> cm to the right of its resting place when <code class='latex inline'>f(t) = 8</code>, and <code class='latex inline'>8</code> cm to the left of its resting place when <code class='latex inline'>f(t) = -8</code>. If the flagpole is observed for <code class='latex inline'>2</code>s, it appears to be furthest to the left when <code class='latex inline'>t= 1.5</code> s. Is this observation correct? </p><p>Justify your answer using the appropriate calculations for the rate of change in displacement.</p>
<p>Determine the instantaneous rate of change at <code class='latex inline'>x= 2</code>.</p><p><code class='latex inline'>\displaystyle y =2x - 1</code></p>
<p>For the function <code class='latex inline'>f(x) = 6x^2 - 4</code>, what is an estimate of the instantaneous rate of change for the given value of <code class='latex inline'>x</code>?</p><p><code class='latex inline'>x = -2</code></p>
<p>Which interval gives the best estimate of the tangent at <code class='latex inline'>x = 3</code> on a smooth curve?</p> <ul> <li><strong>A.</strong> <code class='latex inline'>2 < x < 4</code></li> <li><strong>B.</strong> <code class='latex inline'>2 < x < 3</code></li> <li><strong>C.</strong> <code class='latex inline'>3 < x < 3.3</code></li> <li><strong>D.</strong> cannot be sure</li> </ul>
<p>What is an estimate the slope of the tangent line when <code class='latex inline'>x = 2</code>?</p><p><code class='latex inline'> f(x) = 3^x + 1 </code></p>
<p>An object is sent through the air. Its height is modelled by the function <code class='latex inline'>h(x) = -5x^2 + 3x + 65</code>, where <code class='latex inline'>h(x)</code> is the height of the object in metres and <code class='latex inline'>x</code> is the time in seconds. What is an estimate of the instantaneous rate of change in the object&#39;s height at <code class='latex inline'>3</code> s?</p>
<p>At <code class='latex inline'>\displaystyle x=5 </code>, the function <code class='latex inline'>\displaystyle f(x)=13 x-1.3 x^{2}+7.3 </code> has a) a maximum</p><p>b) a minimum</p><p>c) both a maximum and a minimum</p><p>d) neither a maximum nor a minimum</p>
<p>a) Find the maximum and minimum values for each exponential growth or decay equation on the given interval.</p><p>i) <code class='latex inline'>\displaystyle y=100(0.85)^{t} </code>, for <code class='latex inline'>\displaystyle 0 \leq t \leq 5 </code></p><p>ii) <code class='latex inline'>\displaystyle y=35(1.15)^{x} </code>, for <code class='latex inline'>\displaystyle 0 \leq x \leq 10 </code></p><p>b) Examine your answers for part a). Use your answers to hypothesize about where the maximum value will occur in a given range of values, <code class='latex inline'>\displaystyle a \leq x \leq b . </code> Explain and support your hypothesis thoroughly.</p><p>a) Find the maximum and minimum values for each exponential growth or decay equation on the given interval.</p><p>i) <code class='latex inline'>\displaystyle y=100(0.85)^{t} </code>, for <code class='latex inline'>\displaystyle 0 \leq t \leq 5 </code></p><p>ii) <code class='latex inline'>\displaystyle y=35(1.15)^{x} </code>, for <code class='latex inline'>\displaystyle 0 \leq x \leq 10 </code></p><p>b) Examine your answers for part a). Use your answers to hypothesize about where the maximum value will occur in a given range of values, <code class='latex inline'>\displaystyle a \leq x \leq b . </code> Explain and support your hypothesis thoroughly.</p>
<p>Consider the function <code class='latex inline'>f(x)=3(x-2)^2-2</code>.</p><p>Estimate the instantaneous rate of change at <code class='latex inline'>x=4</code>.</p>
<p>a) For the function <code class='latex inline'>\displaystyle y=3 x^{2}+2 x </code>, use a simplified algebraic expression in terms of <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle h </code> to estimate the slope of the tangent at each of the following <code class='latex inline'>\displaystyle x </code>-values, when <code class='latex inline'>\displaystyle b=0.1, b=0.01 </code>, and <code class='latex inline'>\displaystyle h=0.001 </code>.</p><p>i) <code class='latex inline'>\displaystyle a=2 </code></p><p>ii) <code class='latex inline'>\displaystyle a=-3 </code></p><p>b) Determine the equation of the tangent at the above <code class='latex inline'>\displaystyle x </code>-values.</p><p>c) Graph the curve and tangents.</p>
<p>If a ball is thrown into the air with a velocity of <code class='latex inline'>40 ft/s</code>, its height (in feet) after t seconds is given by <code class='latex inline'>y = 40t - 16t^2</code>. Find the velocity when <code class='latex inline'> t = 2</code> You may use power rule for this.</p>
<p>For each function, the point given is the maximum or minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero.</p><p><code class='latex inline'>\displaystyle f(x) = -4.5\cos(2x); </code> <code class='latex inline'>\displaystyle (0^o, -4.5) </code></p>
<ul> <li><strong>(a)</strong> State the coordinates of the tangent point.</li> <li><strong>(b)</strong> State the coordinates of another point on the tangent line.</li> <li><strong>(c)</strong> Use the points you found in parts a) and b) to determine the slope of the tangent line.</li> </ul> <img src="/qimages/305" />
<p>Find the slope of the tangent line when <code class='latex inline'>x = 2</code>.</p><p><code class='latex inline'> f(x) = 3x^2 - 5x + 1 </code></p><p>Show your work.</p>
<p>For the growth equation <code class='latex inline'>\displaystyle y=35(1.7)^{x} </code>, the maximum value over the domain <code class='latex inline'>\displaystyle 0 \leq x \leq 8 </code> is <code class='latex inline'>\displaystyle \begin{array}{ll}\text { a) } y=1.7 & \text { c) } y=69.7 \\ \text { b) } y=2441.5 & \text { d) } y=35\end{array} </code></p>
<img src="/qimages/14205" /><p>For the following graph, what is the best estimate of the slope of the tangent at <code class='latex inline'>\displaystyle x=2 ? </code> a) 2</p><p>b) <code class='latex inline'>\displaystyle -1 </code></p><p>c) <code class='latex inline'>\displaystyle -2 </code></p><p>d) <code class='latex inline'>\displaystyle -3 </code></p>
<p>At each of the indicated points on the graph, is the instantaneous rate of change positive, negative, or zero? Explain.</p><img src="/qimages/306" />
<p>Consider the function <code class='latex inline'>f(x)=3(x-2)^2-2</code>.</p><p>State the coordinates of a point where the instantaneous rate of change in <code class='latex inline'>f(x)</code> will be negative.</p>
<p>When polluted water begins to flow into an unpolluted pond, the concentration of pollutant, <code class='latex inline'>c</code>, in the pond at <code class='latex inline'>t</code> minutes is modelled by <code class='latex inline'>c(t)=\displaystyle{\frac{27t}{10000+3t}}</code>, where <code class='latex inline'>c</code> is measured in kilograms per cubic metre. Determine the rate at which the concentration is changing after</p> <ul> <li>one week</li> </ul>
<p>A construction worker drops a bolt while working on a high-rise building 320 m above the ground. After <code class='latex inline'>t</code> seconds, the bolt&#39;s height above the ground is s meters, where <code class='latex inline'>s(t)=320-5t^2, 0 \leq t \leq 8</code>.</p><p>Find the average velocity for the interval <code class='latex inline'>3 \leq t \leq 8</code>.</p><p>Find the bolt&#39;s velocity at <code class='latex inline'>t = 2</code>.</p>
<p>For the graph shown, estimate the slope of the tangent line at <code class='latex inline'>(7, 5)</code>.</p><img src="/qimages/1693" />
<p>A population of raccoons moves into a wooded area. At <code class='latex inline'>t</code> months, the number of raccoons, P(t), can be modeled using the equation <code class='latex inline'>P(t) = 100 + 30t + 4t^2</code>.</p><p>What is an estimate of the rate of change in the raccoon population at exactly <code class='latex inline'>2.5</code> months?</p>
<p>Find the slope of the tangent line when <code class='latex inline'>x = 2</code>.</p><p><code class='latex inline'> f(x) = 3^x + 1 </code></p><p>Show your work.</p>
<p>The graph of a rational function is shown</p><img src="/qimages/486" /><p>Determine the slope of the tangent line to estimate the instantaneous rate of change at this point.</p>
<p>A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.</p><p><code class='latex inline'>g(x) = -x^2 -6x - 4; (-3, 5)</code></p>
<p>An oil tank is being drained. The volume, <code class='latex inline'>V</code>, in litres, of oil remaining in the tank after time, <code class='latex inline'>t</code>, in minutes, is represented by the function <code class='latex inline'>V(t) = 60(25 - t)^2</code>, <code class='latex inline'>0\leq t \leq 25</code>.</p><p>Estimate the instantaneous rate of change of the volume at each of the following times. </p> <ul> <li>i) t = 5</li> <li>ii) t = 10</li> <li>iii) t = 15</li> <li>iv) t = 20</li> </ul> <p>Compare these values, giving reasons for any similarities and differences. </p>
<p>Use an algebraic strategy to verify that the point given for the function is either a maximum or minimum.</p><p><code class='latex inline'>\displaystyle f(x)=x^{3}-3 x ;(-1,2) </code></p>
<p>A soccer ball is kicked into the air such that its height, <code class='latex inline'>h</code>, in metres, after t seconds can be modelled by the function <code class='latex inline'>h(t) = -4.9t^2 +12t + 0.5.</code></p><p>Determine the average rate of change of the height of the ball from 1 s to 2 s.</p>
<p>When polluted water begins to flow into an unpolluted pond, the concentration of pollutant, <code class='latex inline'>c</code>, in the pond at <code class='latex inline'>t</code> minutes is modelled by <code class='latex inline'>c(t)=\displaystyle{\frac{27y}{10000+3t}}</code>, where <code class='latex inline'>c</code> is measured in kilograms per cubic metre. Determine the rate at which the concentration is changing after</p> <ul> <li><code class='latex inline'>1 h</code></li> </ul>
<p>The demand function for snack cakes at a large bakery is given by the function <code class='latex inline'>C(x)=\displaystyle{\frac{x^2-4x+20}{x}}</code>, where <code class='latex inline'>x</code> is the number of T-shirts produced, in thousands. <code class='latex inline'>C(x)</code> is measured in dollars.</p><p>Calculate the average cost of a T-shirt at a production level of 3000 pairs.</p>
<p>Find the slope of the tangent line when <code class='latex inline'>x = 2</code>.</p><p><code class='latex inline'> f(x) = \sqrt{x + 2} </code></p><p>Show your work.</p>
<p>Table of values are from the function <code class='latex inline'>f(x) = 5x^2 - 7</code> below.</p><img src="/qimages/119" /><p>Based on the trend in the average rates of change, what is an estimate of the instantaneous rate of change when <code class='latex inline'>x = 2</code>?</p>
<p>The population, <code class='latex inline'>P</code>, of a small town after t years can be modelled by the function <code class='latex inline'>P(t) = 0.5t^3+150t + 1200</code>, where <code class='latex inline'>t = 0</code> represents the beginning of this year.</p> <ul> <li>Write an expression for the average rate of change of the population from <code class='latex inline'>t = 8</code> to <code class='latex inline'>t = 8 + h</code>.</li> </ul>
<p>Determine the instantaneous rate of change at <code class='latex inline'>x= 2</code>.</p><p><code class='latex inline'>\displaystyle y =x - 2x^3</code></p>
<p>The data show the percent of households that play games over the Internet.</p><img src="/qimages/308" /> <ul> <li>Determine the average rate of change, in percent, of households that played games over the internet from 1999 to 2003.</li> </ul>
<p>The graph shows the amount of water remaining in a pool after it has been draining for <code class='latex inline'>\displaystyle 4 \mathrm{~h} </code></p><p>Volume of Water Remaining in Pool</p><img src="/qimages/154783" /><p>a) What does the graph tell you about the rate at which the water is draining? Explain.</p><p>b) Determine the average rate of change of the volume of water remaining in the pool during the following intervals.</p> <ul> <li>i) the first hour</li> <li>ii) the last hour</li> </ul> <p>c) Determine the instantaneous rate of change of the volume of water remaining in the pool at each time.</p> <ul> <li>i) <code class='latex inline'>\displaystyle 30 \mathrm{~min} </code></li> <li>ii) <code class='latex inline'>\displaystyle 1.5 \mathrm{~h} </code></li> <li>iii) <code class='latex inline'>\displaystyle 3 \mathrm{~h} </code></li> </ul> <p>d) How would this graph change under the following conditions? Justify your answer.</p> <ul> <li>i) The water was draining more quickly.</li> <li>ii) There was more water in the pool at the beginning.</li> </ul> <p>e) Sketch a graph of the instantaneous rate of change of the volume of water remaining in the pool versus time for the graph shown.</p>
<p>Select a strategy to estimate the instantaneous rate of change of each function at the given point.</p><p><code class='latex inline'>y=\displaystyle{\frac{-3x^2+5x+6}{x+6}}</code>, where <code class='latex inline'>x=-3</code></p>
<p>An offshore oil platform develops a leak. As the oil spreads over the surface of the ocean, it forms a circular pattern with a radius that increases by 1 m every 30 s.</p><p>a) Construct a table of values that shows the area of the oil spill at 2—min intervals for 30 min, and graph the data.</p><p>b) Determine the average rate of change of the area during each interval.</p> <ul> <li>i) the first 4 min</li> <li>ii) the next 10 min</li> <li>iii) the entire 30 min</li> </ul> <p>c) What is the difference between the instantaneous rate of change of the area of the spill at 5 min and at 25 min?</p>
<p>A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.</p><p><code class='latex inline'>f(x) =x^2 -10x + 7; (5, -18)</code></p>
<p>For the graph shown, estimate the slope of the tangent line at <code class='latex inline'>(4, 2)</code>.</p><img src="/qimages/1693" />
<p>Given the function <code class='latex inline'> \displaystyle f(x) = \frac{2x}{x -4} </code>, determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through <code class='latex inline'>A(5, 10)</code> and <code class='latex inline'>B(8, 4)</code>.</p>
<p><strong>a)</strong> For <code class='latex inline'>f(x)</code>, find the equation for the slope of the secant line between any general point on the function <code class='latex inline'>(a + h, f(a + h))</code> and the given x-coordinate of another point.</p> <ul> <li><code class='latex inline'>f(x) = x^2 -30x; a = 2</code></li> </ul> <p><strong>b)</strong> Use each slope equation you found in part a) to estimate the slope of the tangent line at the point with the given x—coordinate.</p>
<p>For the graph shown, estimate the slope of the tangent line at <code class='latex inline'>(5, 1)</code>.</p><img src="/qimages/1693" />
<p>For each function, the point given is the maximum of minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero. </p><p><code class='latex inline'>f(x) = 0.5x^2 + 6x + 7.5; (-6, -10.5)</code></p>
<p>Select a strategy to estimate the instantaneous rate of change of each function at the given point.</p><p><code class='latex inline'>y=\displaystyle{\frac{1}{25-x}}</code>, where <code class='latex inline'>x=13</code></p>
<p>The data show the percent of households that play games over the Internet.</p><img src="/qimages/308" /> <ul> <li>i. in 2000</li> <li>ii. in 2002</li> </ul>
<p>What are the slopes of these secant lines? What is an estimate the slope of the tangent line when <code class='latex inline'>x = 2</code>?</p><p><code class='latex inline'> f(x) = \sqrt{x + 2} </code></p>
<p>The population, <code class='latex inline'>P</code>, of a small town after <code class='latex inline'>t</code> years can be modelled by the function <code class='latex inline'>P(t) = 0.5t^3+150t + 1200</code>, where <code class='latex inline'>t = 0</code> represents the beginning of this year.</p><p>Determine the average rate of change of the population when</p><p><code class='latex inline'> \displaystyle \begin{array}{cccccc} &(i)& h = 2 &(ii)& h = 4 &(iii)& h = 5 \end{array} </code></p>
<p>For the function <code class='latex inline'>f(x) = 6x^2 - 4</code>, what is an estimate of the instantaneous rate of change for the given value of <code class='latex inline'>x</code>?</p><p><code class='latex inline'>x = 8</code></p>
<p>A pilot who is flying at an altitude of <code class='latex inline'>10 000</code> ft is forced to eject from his airplane. The path his ejection seat takes is modelled by the equation <code class='latex inline'>h(t) = -16t^2 + 90t + 10000</code>, where <code class='latex inline'>h(t)</code> is his altitude in feet and t is the time since his ejection in seconds. estimate at what time, <code class='latex inline'>t</code>, the pilot is at a maximum altitude. Explain how the maximum altitude is related to the slope of the target line at certain points.</p>
<p>Estimate the instantaneous rate of change at the tangent point indicated on each graph.</p><img src="/qimages/7144" />
<p>Determine the slope of the line that is tangent to the graph of each function at the given point. Then determine the value of <code class='latex inline'>x</code> at which there is no tangent line.</p><p><code class='latex inline'>f(x)=\displaystyle{\frac{2x^2-6x}{3x+5}}</code>, where <code class='latex inline'>x=-2</code></p>
<p>Write difference quotient that can be used to estimate the slope of the tangent to the function <code class='latex inline'>y =f(x)</code> at <code class='latex inline'>x = 4</code></p>
<p>A family purchased a home for <code class='latex inline'>\$125 000</code>. Appreciation of the home&#39;s value, in dollars, can be modelled by the equation <code class='latex inline'>H(t) = 125 000(1.06)^t</code>, where <code class='latex inline'>H(t)</code> is the value of the home and <code class='latex inline'>t</code> is the number of years that the family owns the home . What is an estimate of the instantaneous rate of change in the home&#39;s value at the start of the eight year of owning the home?</p>
<p>A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.</p><p><code class='latex inline'>k(x) = -4\cos(x + 25); (-25^o, -4)</code></p>
<p>The height of a diver above the water is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+5 t+10 </code>, where <code class='latex inline'>\displaystyle t </code> represents the time in seconds and <code class='latex inline'>\displaystyle h(t) </code> represents the height in metres. Use the appropriate calculations for the rate of change in height to show that the diver reaches her maximum height at <code class='latex inline'>\displaystyle t=0.5 \mathrm{~s} </code></p><p>The height of a diver above the water is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+5 t+10 </code>, where <code class='latex inline'>\displaystyle t </code> represents the time in seconds and <code class='latex inline'>\displaystyle h(t) </code> represents the height in metres. Use the appropriate calculations for the rate of change in height to show that the diver reaches her maximum height at <code class='latex inline'>\displaystyle t=0.5 \mathrm{~s} </code></p>
<p>The demand function for snack cakes at a large bakery is given by the function <code class='latex inline'>C(x)=\displaystyle{\frac{x^2-4x+20}{x}}</code>, where <code class='latex inline'>x</code> is the number of T-shirts produced, in thousands. <code class='latex inline'>C(x)</code> is measured in dollars.</p><p>Estimate the rate at which the average cost is changing at a production level of 3000 T-shirts.</p>
<p>The height, in centimetres, of a piston attached to a turning wheel at time t, in seconds, is modelled by the equation <code class='latex inline'>y = 2 \sin (120^ot)</code>.</p><p>Find the instantaneous rate of change at <code class='latex inline'>t = 12</code> s.</p>
<p>Compare each of the following expressions to the different quotient <code class='latex inline'> \displaystyle \frac{f(a + h)-f(a)}{h} </code>, identifying</p> <ul> <li>i) the equation of <code class='latex inline'>y=f(x)</code></li> <li>ii) the value of a</li> <li>iii) the value of h</li> <li>iv) the tangent point (a, f(a))</li> </ul> <p><code class='latex inline'> \displaystyle \frac{4.01^2 -16}{0.01} </code></p>
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