Q: Which of the following is the correct derivative of $$y = \tan(\sin \theta)$$ with respect to $$\theta$$?

$$\sec^2$(\sin \theta) \cdot \cos \theta$$

Q: Use the Mean Value Theorem to determine if $$f(x) = 7 - x^2$ on $$[-1,2]$$ satisfies the hypotheses. If so, what value(s) of c$ in (-1$$,2)$$ satisfy the theorem?

$$c = -0.5$$ \text{ and } c = 0.5$

Q: Evaluate $$\displaystyle \$$int_0$$^{\pi/4} \sec^2$ x\,dx.$$

$$\tan\left(\frac{\pi}{4}\right) - \tan(0)$$

Question 1

An object moves along a straight line with position function $$s(t) = t^2$ - 12t$$ for $$0 \leq t \leq 13. At $$what time(s) is the object stationary?

Accepted Answer

$$t = 0$$ 	ext{ and } t = 12$

Question 2

Given $$f(x) = x e$$^{-10x}$$, what is f$$'(x)$$?

Accepted Answer

$$e^{-10x} - 10x$$ $$e^{-10x}$$

Question 3

Evaluate the limit: $$\displaystyle \lim_{x 	o -\infty} x^3$.

Accepted Answer

$$-\infty$$

Question 4

Find $$\displaystyle \frac{dy}{dx}$$ for the curve $$x + y = \cos(2x).$$

Accepted Answer

$$\displaystyle \frac{dy}{dx} = -1 + 2\sin(2x)$$

Question 5

Which of the following is the correct derivative of $$y = 	an(\sin 	heta)$$ with respect to $$	heta$$?

Accepted Answer

$$\sec^2$(\sin 	heta) \cdot \cos 	heta$$

Question 6

Evaluate the definite integral $$\displaystyle \$$int_0$$^2 (3x^2$ + 2x)\,dx.$$

Accepted Answer

$$14$$

Question 7

Suppose $$f(x) = 2x^5$ - 5x^4$ - 10x^3$ + 4 on$$ $$[-2,4]. $$Which of the following is a critical point of $$f(x)$$?

Accepted Answer

$$x = -1$$

Question 8

Use the Mean Value Theorem to determine if $$f(x) = 7 - x^2$ on $$[-1,2]$$ satisfies the hypotheses. If so, what value(s) of c$ in (-1$$,2)$$ satisfy the theorem?

Accepted Answer

$$c = -0.5$$ 	ext{ and } c = 0.5$

Question 9

Evaluate $$\displaystyle \$$int_0$$^{\ln 8} e^x\,dx.$$

Accepted Answer

$$8$$

Question 10

Given $$a(t) = -32$$, $$v(0) = 70$$, and $$s(0) = 10$$, what is the position function $$s(t)$$ for a projectile?

Accepted Answer

$$s(t) = -32t^2$ + 70t + 10$$

Question 11

Evaluate the limit using l'Hôpital's Rule: $$\displaystyle \lim_{x 	o 3} \frac{\ln x}{4x - x^2$ - 3}.$$

Accepted Answer

$$\frac{1}{-6}$$

Question 12

Which substitution is most appropriate for evaluating $$\displaystyle \int x^9$ \sin(x$$^{10}$$)\,dx$$?

Accepted Answer

$$u = x$$^{10}$$

Question 13

Calculate the right Riemann sum for $$f(x) = x^3$ - 1 on$$ $$[-2,0]$$ with $$n = 4.$$

Accepted Answer

$$-6$$

Question 14

Find the derivative of $$y = \frac{\ln x}{\ln x + 1}.$$

Accepted Answer

$$\frac{1}{x(\ln x + 1)^2$}$$

Question 15

Evaluate $$\displaystyle \$$int_0$$^{\pi/4} \sec^2$ x\,dx.$$

Accepted Answer

$$	an\left(\frac{\pi}{4}ight) - 	an(0)$$

Question 16

For the position function $$s(t) = t^2$ - 12t$$, what is the acceleration at $$t = 1$$?

Accepted Answer

$$2$$

Question 17

Find the equation of the tangent line to $$f(x) = \frac{1}{2x+1} at$$ $$P(0,1).$$

Accepted Answer

$$y = -2x + 1$$

Calculus I Final Exam Review – Step-by-Step Guidance

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