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Similar Question 1
<p>For this graph of a function, state two points where the function has an instantaneous rate of change in <code class='latex inline'>f(x)</code> that is</p><p>c) a positive value</p><img src="/qimages/502" />
Similar Question 2
<p>For <code class='latex inline'>f(x) = \cos x</code> on <code class='latex inline'>x\in [0, 2\pi]</code>.</p><p><strong>(a)</strong> For what values of <code class='latex inline'>x</code> does the instantaneous rate of change appear to equal 0?</p><p><strong>(b)</strong> For what values of <code class='latex inline'>x</code> does the instantaneous rate of change appear to reach a maximum value? a minimum value?</p>
Similar Question 3
<p>The durations of daylight in Sarnia, Ontario, on the first of the month from January to December are shown.</p><img src="/qimages/2497" /><p><strong>(a)</strong> Copy the table. Add a column to express the durations of daylight as decimal values, to the nearest hundredth of an hour.</p><p><strong>(b)</strong> Write a sine function to model the data.</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
Now You Try
<p>Estimate the instantaneous rate of change at <code class='latex inline'>x = 2</code> for each.</p><p><strong>(a)</strong> <code class='latex inline'> \displaystyle y = x </code></p><p><strong>(b)</strong> <code class='latex inline'> \displaystyle y = x^2 </code></p><p><strong>(c)</strong> <code class='latex inline'> \displaystyle y = x^3 </code></p><p><strong>(d)</strong> <code class='latex inline'> \displaystyle y = 7 </code></p>
<p><strong>(a)</strong> Sketch a graph of the tangent function on the interval <code class='latex inline'>x\in</code> <code class='latex inline'>[0,2\pi]</code>.</p><p><strong>(b)</strong> For what values of <code class='latex inline'>x</code> does the instantaneous rate of change appear to equal 0? reach a maximum value? reach a minimum value?</p>
<p>For the following graph of a function, state two intervals in which the function has an average rate of change in <code class='latex inline'>f(x)</code> that is</p><p>a) zero</p><img src="/qimages/501" />
<p>For this graph of a function, state two points where the function has an instantaneous rate of change in <code class='latex inline'>f(x)</code> that is</p><p>c) a positive value</p><img src="/qimages/502" />
<p>For this graph of a function, state two points where the function has an instantaneous rate of change in <code class='latex inline'>f(x)</code> that is</p><p>a) zero</p><img src="/qimages/502" />
<p>For the following graph of a function, state two intervals in which the function has an average rate of change in <code class='latex inline'>f(x)</code> that is</p><p>b) a negative value</p><img src="/qimages/501" />
<p>For this graph of a function, state two points where the function has an instantaneous rate of change in <code class='latex inline'>f(x)</code> that is</p><p>b) a negative value</p><img src="/qimages/502" />
<p>For the following graph of a function, state two intervals in which the function has an average rate of change in <code class='latex inline'>f(x)</code> that is</p><p>c) a positive value</p><img src="/qimages/501" />
<p><strong>(a)</strong> Sketch a graph of the cosecant function on the interval <code class='latex inline'>x \in [0,2\pi]</code>.</p><p><strong>(b)</strong> For what values of <code class='latex inline'>x</code> does the instantaneous rate of change appear to equal 0? reach a maximum value? reach a minimum value?</p>
<p>A rollercoaster at a theme park starts with a vertical drop that leads into two pairs of identical valleys and hills, as shown.</p><img src="/qimages/2169" /><p><strong>(a)</strong> Write a quadratic function to model the first section of the rollercoaster.</p><p><strong>(b)</strong> Write a sinusoidal function to model the second section of the rollercoaster.</p>
<p>The height, <code class='latex inline'>h</code>, in metres, of a car above the ground as a Ferris wheel turns can be modelled using the function <code class='latex inline'>\displaystyle{h=15\cos\left(\frac{\pi t}{120}\right)+18}</code>, where <code class='latex inline'>t</code> is the time, in seconds.</p><p><strong>(a)</strong> Determine the average rate of change of <code class='latex inline'>h</code> in the following time intervals, rounded to three decimal places.</p> <ul> <li>i) 15 s to 20 s </li> <li>ii) 19 s to 20 s</li> <li>iii) 19.9 s to 20 s</li> <li>iv) 19.99 s to 20 s</li> </ul> <p><strong>(b)</strong> Estimate a value for the instantaneous rate of change of <code class='latex inline'>h</code> at <code class='latex inline'>t=20</code> s.</p>
<p>A weight is suspended on a spring and set in motion such that it bobs up and down vertically. The graph shows the height, <code class='latex inline'>h</code>, in centimetres, of the weight above a desk after time, <code class='latex inline'>t</code>, in seconds. Use the graph to determine a model of the height versus time using a cosine function.</p><img src="/qimages/2167" />
<img src="/qimages/2169" /><p>(a) Determine the instantaneous rate of change of the function that you used to model the first section of the rollercoaster at <code class='latex inline'>x=2</code></p><p>(b) Determine the instantaneous rate of change of the function that you used to model the second section of the rollercoaster at <code class='latex inline'>x=2</code></p><p>(c) How do the answers to parts (a) and (b) compare?</p><p>(Question 11)</p><p>A rollercoaster at a theme park starts with a vertical drop that leads into two pairs of identical valleys and hills, as shown.</p><img src="/qimages/2169" /><p><strong>(a)</strong> Determine the instantaneous rate of change of the function that you used to model the first section of the rollercoaster at <code class='latex inline'>x = 2</code>,</p><p><strong>(b)</strong> Determine the instantaneous rate of change of the function that you used to model the second section of the rollercoaster at <code class='latex inline'>x = 2</code>,</p><p><strong>(c)</strong> How do the answers to parts a) and b) compare?</p>
<p><strong>(a)</strong> Sketch a graph of the inverse trigonometric relation <code class='latex inline'>y=\sin^{-1}(x)</code> such that the range covers the interval <code class='latex inline'>[0,2\pi]</code>.</p><p><strong>(b)</strong> Is this relation a function in this range? If so, explain how you know. If not, show how it can be made into a function by restricting the range.</p><p><strong>(c)</strong> Determine a value of <code class='latex inline'>x</code> where the instantaneous rate of change appears to be a maximum.</p><p><strong>(d)</strong> Estimate the instantaneous rate of change for the value of <code class='latex inline'>x</code> in part (c).</p>
<p>For <code class='latex inline'>f(x) = \cos x</code> on <code class='latex inline'>x\in [0, 2\pi]</code>.</p><p><strong>(a)</strong> For what values of <code class='latex inline'>x</code> does the instantaneous rate of change appear to equal 0?</p><p><strong>(b)</strong> For what values of <code class='latex inline'>x</code> does the instantaneous rate of change appear to reach a maximum value? a minimum value?</p>
<p>An antique motorcycle is susceptible to a speed wobble that is modelled by <code class='latex inline'>d = a\sin kt</code>, where <code class='latex inline'>d</code> represents the deviation, in centimetres, of the front of the wheel from forward motion and <code class='latex inline'>t</code> represents the time, in seconds. A graph of deviation versus time is shown.</p><img src="/qimages/2168" /><p><strong>(a)</strong> Use the graph to determine values for <code class='latex inline'>a</code> and <code class='latex inline'>k</code>.</p><p><strong>(b)</strong> Determine an equation for the speed wobble.</p>
<img src="/qimages/2167" /><p><strong>a)</strong> Select a point on the graph where the instantaneous rate of change of the height appears to be a maximum.</p><p><strong>b)</strong> Use <code class='latex inline'> \displaystyle Av Rate of Change = \frac{h_2-h_1}{t_2-t_1} </code> to estimate the instantaneous rate of change of the height at this point.</p><p><strong>c)</strong> What does this instantaneous rate of change of the height represent?</p>
<p>The durations of daylight in Sarnia, Ontario, on the first of the month from January to December are shown.</p><img src="/qimages/2497" /><p><strong>(a)</strong> Copy the table. Add a column to express the durations of daylight as decimal values, to the nearest hundredth of an hour.</p><p><strong>(b)</strong> Write a sine function to model the data.</p>
<p>Use the graph to calculate the average rate of change in <code class='latex inline'>f(x)</code> on the interval <code class='latex inline'>2 \leq x \leq 5</code>.</p><img src="/qimages/503" />
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