Question 1

Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 4$$ seconds?

Accepted Answer

$$32 m/s$$

Question 2

Evaluate the following limit using L'Hôpital's Rule: $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 3

Which of the following statements about the continuity of a function $$f(x) at$$ $$x = a is $$FALSE?

Accepted Answer

$$f(x)$$ must be differentiable at $$x = a.$$

Question 4

Find the equation of the tangent line to the curve $$y = e$$^{2x}$$ at the point where x = 0$.

Accepted Answer

$$y = 2x + 1$$

Question 5

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V(t) = 5t^3$. What is the rate at which water is being added to the tank at t = 2$ minutes?

Accepted Answer

$$60 L/$$min$

Question 6

Evaluate $$\displaystyle \$$int_0$$^1 (3x^2$ + 2x)\,dx.$$

Accepted Answer

$$2.5$$

Question 7

Which of the following is the correct derivative of $$y = \ln(x^2$ + 1)$$?

Accepted Answer

$$\displaystyle \frac{2x}{x^2$ + 1}$$

Question 8

A company wants to design a rectangular enclosure with three sides using fencing (the fourth side is a river and needs no fence). If $$200$$ meters of fencing are available, what is the maximum area that can be enclosed?

Accepted Answer

$$5000 m$^2$$

Question 9

Find the value of $$c$$ guaranteed by the Mean Value Theorem for $$f(x) = x^2$ on $$[1, 5]$$.

Accepted Answer

$$c = 3$$

Question 10

Which of the following integrals represents the area between the curves $$y = x^2$ and y = x + 2$ from x = 0$ to x = 2$?

Accepted Answer

$$\displaystyle \$$int_0$$^2 [(x + 2) - x^2$] \, dx$$

Question 11

If $$f(x) = \sin(x)$$, what is the second derivative $$f''(x)$$?

Accepted Answer

$$-\sin(x)$$

Question 12

Which of the following is the correct application of the chain rule for $$\displaystyle \frac{d}{dx} \left[ \sin(3x^2$) \right]$$?

Accepted Answer

$$6x \cos(3x^2$)$$

Question 13

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 14

Given $$y^2$$ + $$x^2 = 25$, find $$\displaystyle \frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\displaystyle -\frac{x}{y}$$

Question 15

A ball is thrown vertically upward with an initial velocity of $$20 m/s $$from a height of $$50 m. $$The position function is $$s(t) = -4.9t^2$ + 20t + 50. $$What is the maximum height reached by the ball?

Accepted Answer

$$70.41 m$$

Question 16

Find $$\displaystyle \int \frac{1}{x^2$ + 4} \, dx.$$

Accepted Answer

$$\displaystyle \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C$$

Question 17

Which of the following is the correct linearization of $$f(x) = \sqrt{x} at$$ $$x = 4$$?

Accepted Answer

$$L(x) = 2 + \frac{1}{4}(x - 4)$$

Question 18

Evaluate $$\displaystyle \$$int_0$$^\pi \sin(x) \, dx.$$

Accepted Answer

$$2$$

Question 19

Which of the following is the correct formula for Newton's Method for approximating a root of $$f(x)$$?

Accepted Answer

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Comprehensive Study Guide: Limits, Derivatives, and Integrals in Calculus

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