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CLEAR Calculus

Chain Rule

You will be asked to find the derivatives of multiple functions using the chain rule. You will need to use the provided virtual maninipulative to gather the derivative information about the functions \(f(x)\) and \(g(y)\). 

The first few questions will orient you to the virtual manipulative and the functions \(f\) and \(g\).

Controlling the Virtual Manipulative

Within the virtual environment you can control which function is displayed, the input value of the displayed function, the scale at which changes to the function and its input are displayed, and the amout of zoom on both sides of the display. 

The left screen of the display shows an interval along the function's input axis (either \(x\) or \(y\)), centered at the input value specified in the input box on the top right of the screen. The right screen shows an interval of the function's output axis (either \(f(x)\) or \(g(y)\)), centered at the corresponding output value of the function. 

The red arrows indicate a change in the forward direction, and the red tiks mark the boundaries of the corresponding intervals of change in that same direction. The blue tiks mark the boundaries of intervals of change in the opposite direction. 

The leftmost slider controls the scale at which changes to the function's inputs are measured, and this similarly determines the corresponding intervals of the function's output on the right screen. The "Zoom" sliders independently control the number of change intervals you view at one time on either screen. 

The check boxes at the top right of the screen control which function is displayed.


Click ⛶ to open in full screen

Warm Up Questions

  1. Select the "Use \(f\)" checkbox. Set the input to \(x=5\). Adjust the scale so that the intervals of change in \(x\) on the left screen correspond to a change amount of \(\Delta x\) between \(.1\) and \(.2\). Which of the following describe the corresponding sizes of the displayed intervals of change in \(f(x)\) on the right screen? (Select all that apply)

    1. The intervals of change \(\Delta f\) are all twice as large as \(\Delta x\) and oriented in the same direction as \(\Delta x\).

    2. The intervals of change \(\Delta f\) get smaller as \(x\) gets bigger.

    3. All intervals of change \(\Delta f\) are the same size.

    4. The intervals of change \(\Delta f\) are all twice as large as \(\Delta x\) and oriented in the opposite direction as \(\Delta x\).

    5. The intervals of change \(\Delta f\) get larger as \(x\) gets bigger.

    6. The intervals of change \(\Delta f\) are all four times as large as \(\Delta x\) and oriented in the same direction as \(\Delta x\).

    7. All intervals of change \(\Delta f\) are the same size as \(\Delta x\).

    8. The intervals of change \(\Delta f\) are all four times as large as \(\Delta x\) and oriented in the opposite direction as \(\Delta x\).

  2. Select the "Use \(g\)" checkbox. Set the input to \(y=1.5\). Adjust the scale so taht the intervals of change in \(y\) are at most of size \(\Delta y=.4\), and zoom out as much as possible on both windows. Which of the following describe the corresponding sizes of the displayed intervals of change in \(g(y)\) on the right screen? (Select all that apply)

    1. The intervals of change \(\Delta g\) are all twice as large as \(\Delta y\) and oriented in the same direction as \(\Delta y\).

    2. The intervals of change \(\Delta g\) get smaller as \(y\) gets bigger.

    3. All intervals of change \(\Delta g\) are the same size.

    4. The intervals of change \(\Delta g\) are all three times as large as \(\Delta y\) and oriented in the opposite direction as \(\Delta y\).

    5. The intervals of change \(\Delta g\) get larger as \(y\) gets bigger.

    6. All intervals of change \(\Delta g\) are the same size as \(\Delta y\).

    7. The intervals of change \(\Delta g\) are all four times as large as \(\Delta y\) and oriented in the opposite direction as \(\Delta y\).

  3. Keep \(g\) active and \(y=1.5\). As you decrease the size of \(\Delta y\), which best describes the resulting relationship between the intervals in \(\Delta g\) near \(g(1.5)\) on the right screen? (Zoom in and out as needed)

    1. The intervals of change \(\Delta g\) become less evenly spaced for small values of \(\Delta y\).

    2. The intervals of change \(\Delta g\) become more evenly spaced for small values of \(\Delta y\).

    3. The intervals of change \(\Delta g\) maintain their relative spacing for all values of \(\Delta y\).

  4. Adjust \(\Delta y\) until it has a size of at most \(.1\), and zoom in near \(g(1.5)\). Using the amounts of change \(\Delta g\), estimate \(g'(1.5)\).

The Chain Rule

  1. Our first goal is to find \(h'(1.5)\), where \(h(y)=f(g(y))\). To estiamte \(h'(1.5)\), we first must estiamte \(f'(x)\) for a particular values of \(x\). Which adequatly describe the desired value of \(x\)? (Select all that apply).

    1. \(x=f(1.5)\)

    2. \(x=2.25\)

    3. \(x=1.5\)

    4. \(x=-3.25\)

    5. \(x=g(1.5)\)

    6. \(x=h(1.5)\)

  2. Estimate \(f'(x)\) for the value of \(x\) you selected in the previous question.

  3. Use your results from the previous two questions to estimate \(h'(1.5)\).

  4. Use similar reasoning to estimate \(r'\left(\frac{\pi}{4}\right)\) where \(r(\theta)=g(\sin(\theta))\).

  5. Use similar reasoning to estimate \(s'(1)\) where \(s(x)=\sin(f(x))\).