MCV4U Derivatives of Exponential and Trig Functions Practice Test (with Solutions)

Free Derivatives of Trig, Exp and Log Practice Test

8 questions (15 parts), exactly as a free account gets them in the first Derivatives of Trig, Exp and Log test on Martian Lab. Try each one on paper first, then open the answer.

1. Is the derivative of a sinusoidal function always periodic? Explain why or why not with an example.

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   The derivative of $\sin(x)$ is $\cos(x)$ and the derivative of $\cos(x)$ is $-\sin(x)$. Both derivative functions are periodic. Since all sinusoidal functions can be expressed as $\sin(x)$ or $\cos(x)$, their derivative would also be periodic. $y = \tan x$, $y' = \sec^2 x$ is also periodic.
2. Let $f(x) = b^x$, where $b > 1$, and let $g(x) = \dfrac{f'(x)}{f(x)}$.

   1. Predict the equation and the type of graph of $g(x)$ by giving one example.
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      It would be a constant. $f(x) = 2^x$, $f'(x) = 2^x \cdot \ln 2$ $g(x) = \dfrac{2^x \cdot \ln 2}{2^x} = \ln 2$
   2. For what value of $b$ will $g(x) = e$?
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      $f(x) = b^x$ $f'(x) = b^x \cdot \ln b$ $\dfrac{f'(x)}{f(x)} = \ln b = e$, $b = e^e$.
3. Solve for $x$ and state your solutions in exact value.

   1. $e^{-3x+8} = 12$
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      $-3x+8 = \ln 12$ $3x = 8 - \ln 12$ $x = \dfrac{8 - \ln 12}{3}$
   2. $\ln(\ln(7x-5)) = -4$
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      $\ln(7x-5) = e^{-4}$ $7x-5 = e^{e^{-4}}$ $x = \dfrac{e^{e^{-4}}+5}{7}$
   3. $\ln(3x-2) = -5$
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      $3x-2 = e^{-5}$ $x = \dfrac{e^{-5}+2}{3}$
4. Differentiate but DO NOT simplify.

   1. $y = \sin\!\left(\dfrac{\pi}{3} - 2^x\right)$
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      $y' = \cos\!\left(\dfrac{\pi}{3} - 2^x\right) \cdot \left(-2^x\right) \cdot \ln 2$
   2. $y = \dfrac{\pi^2}{2}\sin\!\left(\dfrac{x}{\pi}\right)$
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      $y' = \dfrac{\pi^2}{2} \cdot \cos\!\left(\dfrac{x}{\pi}\right) \cdot \dfrac{1}{\pi}$
   3. $y = \sqrt{e^{2x} + \sin x}$
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      $y' = \dfrac{1}{2}\!\left(e^{2x}+\sin x\right)^{-\frac{1}{2}} \cdot \left(e^{2x} \cdot 2 + \cos x\right)$
   4. $y = x\sin(4x)$
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      $y' = \sin(4x) + x \cdot \cos(4x) \cdot 4$
   5. Use the Quotient Rule $y = \dfrac{2^x}{\sin^2 x}$
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      $y' = \dfrac{2^x \ln 2 \cdot \sin^2 x - 2^x \cdot 2\sin x \cdot \cos x}{\sin^4 x}$
5. If $f(x) = \dfrac{1}{3}e^{6x} + 4\sin(x)$, find $f''\!\left(\dfrac{\pi}{3}\right)$.

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   $f'(x) = \dfrac{1}{3}e^{6x} \cdot 6 + 4\cos(x) = 2 \cdot e^{6x} + 4\cos(x)$ $f''(x) = 2e^{6x} \cdot 6 - 4\sin(x) = 12e^{6x} - 4\sin(x)$ $f''\!\left(\dfrac{\pi}{3}\right) = 12 \cdot e^{2\pi} - 4 \cdot \sin\!\left(\dfrac{\pi}{3}\right) = 12e^{2\pi} - 2\sqrt{3}$.
6. Given $f(x) = -2\sin(3x)$, determine the equation of the tangent line to the graph of $f(x)$ with Use EXACT radian measure where slope is minimum.

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   $f'(x) = -2\cos(3x) \cdot 3 = -6\cos(3x)$ minimum slope is $-6$, it occurs at $x = 0$. $x = 0$, $y = 0$ $\therefore$ The eq. of tangent is $y = -6x$.
7. Find the equation of the tangent line to the graph of $f(x) = e^{-\frac{x}{2}}$ that is perpendicular to the line $x - 2y - 6 = 0$. Express your answer in EXACT value.

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   $2y = x - 6$, $y = \dfrac{1}{2}x - 3$ $\therefore\; m_\perp = -2$ $f'(x) = e^{-\frac{x}{2}} \cdot \dfrac{1}{2} = -2$ $e^{-\frac{x}{2}} = 4$ $\dfrac{-x}{2} = \ln 4$ $x = -2\ln 4$ $y = e^{-\frac{1}{2}(-2\ln 4)} = 4$ $y = mx + b$ $4 = (-2)(-2\ln 4) + b$ $b = 4 - 4\ln 4$ $\therefore$ The equation of tangent is $y = -2x + (4 - 4\ln 4)$.
8. Show that $f(x) = \dfrac{1-\cos(x)}{\sin(x)}$ has no horizontal tangent within its domain.

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   $f'(x) = \dfrac{\sin x \cdot \sin x - (1-\cos x)\cdot \cos x}{\sin^2 x}$ $= \dfrac{\sin^2 x + \cos^2 x - \cos x}{\sin^2 x}$ $= \dfrac{1 - \cos x}{\sin^2 x}$ For $f'(x) = 0$, $\cos x = 1$. However, when $\cos x = 1$, $\sin x = 0$. It is not within the domain. $\therefore$ There is no horizontal tangent.

This Unit on Martian Lab

The lessons cover the derivatives of logarithmic and exponential functions and their applications, then sinusoidal functions and their applications, and finish with mixed functions.

1. Derivatives of Logarithmic & Exponential Functions
2. Applications of Logarithmic & Exponential Functions
3. Derivatives of Sinusoidal Functions
4. Applications of Sinusoidal Functions
5. Derivatives of Mixed Functions

Then 6 full Derivatives of Trig, Exp and Log practice tests to check yourself.

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