Question 1

A car travels along a straight road. Its position at time $$t (in $$hours) is given by $$s(t) = 4t^3$ - 3t^2$ + 2t (in $$miles). What is the average velocity of the car between $$t = 1$$ and $$t = 3$$ hours?

Accepted Answer

$$28$$ miles/hour

Question 2

Evaluate the limit: $$\displaystyle \lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 3

Which of the following functions is NOT continuous at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{1}{x-2}$$

Question 4

Given $$f(x) = x^2$, use the limit definition of the derivative to find f$$'(3)$$.

Accepted Answer

$$6$$

Question 5

If $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 6

A medicine is injected into a patient and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$ mg/L, where t$ is in hours. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

Question 7

Find $$\frac{d}{dx} \left( \ln(\sin x) \right).$$

Accepted Answer

$$\cot(x)$$

Question 8

Suppose $$y^2$$ + $$x^2 = 25$. Using implicit differentiation, what is $$\frac{dy}{dx}$$ at the point (3$$,4)$$?

Accepted Answer

$$-\frac{4}{3}$$

Question 9

Let $$f(x) = x^3$ - 3x^2$ + 4. $$Find all critical points and classify them using the second derivative test.

Accepted Answer

Critical points at $$x=0 ($$local max), $$x=2 ($$local min)

Question 10

A cylindrical can is to be made to hold $$500 cm$^3 of $$liquid. The material for the sides costs $$0.02$$ per cm$$^2$$ and for the top and bottom $$0.03$$ per cm$$^2. $$What is the radius $$r (in cm) of $$the can that minimizes the cost?

Accepted Answer

$$r = 5.42 cm$$

Question 11

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

Accepted Answer

$$2$$

Question 12

Use L'Hôpital's Rule to evaluate $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 13

Find the average value of $$f(x) = x^2$ on the interval $$[0,2]$$.

Accepted Answer

$$\frac{4}{3}$$

Question 14

Which of the following is the correct statement of the Mean Value Theorem for derivatives?

Accepted Answer

If $$f is $$continuous on $$[a,b]$$ and differentiable on $$(a,b)$$, then there exists $$c \in (a,b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 15

Compute $$\$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 16

A tank is being filled with water at a rate of $$R(t) = 5 + 2t$$ liters per minute. How much water enters the tank in the first $$3$$ minutes?

Accepted Answer

$$24$$ liters

Question 17

Solve the separable differential equation $$\frac{dy}{dx} = 2y$$ with initial condition $$y(0) = 3.$$

Accepted Answer

$$y = 3e$$^{2x}$$

Question 18

Use Newton's Method with initial guess $$x_0 = 1$ to approximate a root of f(x$$) = $$x^2 - 2$ after one iteration.

Accepted Answer

$$x_1 = 1.5$

Question 19

Find the area between the curves $$y = x^2$ and y = x$ on $$[0,1]$$.

Accepted Answer

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

Comprehensive Calculus I Study Guide: Key Topics and Concepts

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