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Similar Question 1
<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>
Similar Question 2
<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" />
Similar Question 3
<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>
Similar Questions
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
Now You Try
<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 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>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>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>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>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>
<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>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>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>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>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>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>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>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'>g(x) = -x^2 -6x - 4; (-3, 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'>h(x) = -2x^2 + 68x + 75; (17, 653)</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'>j(x) = \sin(-2x); (45^o, -1)</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'>t(x)=3x^2-4x+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 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>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>
<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>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>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>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>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 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>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>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>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>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>If a function has a minimum value at <code class='latex inline'>(a, f(a))</code> , what do you know about the slopes of the tangent lines at the following points?</p><p>(a) point to the left of, and very close to, <code class='latex inline'>(a, f(a))</code></p><p>(b) point to the right of, and very close to, <code class='latex inline'>(a, f(a))</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{-3x^2+5x+6}{x+6}}</code>, where <code class='latex inline'>x=-3</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'>f(x) =x^2 -10x + 7; (5, -18)</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>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 graph shown, estimate the slope of the tangent line at <code class='latex inline'>(4, 2)</code>.</p><img src="/qimages/1693" />
<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>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>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>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 xâ€”coordinate.</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>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>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>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>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>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>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>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>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>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>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>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>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>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>
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