Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x = a$$

Question 2

Given $$f(x) = x^3$ - 2x^2$ + x$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2 - 4x + 1$

Question 3

A car's position is given by $$s(t) = 4t^2$ + 2t + 1 (in $$meters, $$t in $$seconds). What is the car's instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$24 m/s$$

Question 4

Which of the following is the correct application of the chain rule for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 5

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

Accepted Answer

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

Question 6

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$ mg/L. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1} mg/L/hr$$

Question 7

Which of the following statements about critical points is TRUE?

Accepted Answer

A critical point occurs where $$f'(x) = 0 or$$ $$f'(x) is $$undefined.

Question 8

Suppose $$f(x) = x^4$ - 4x^2$. Which of the following is a local minimum?

Accepted Answer

$$x = 0 is a $$local maximum, $$x = \pm \sqrt{2}$$ are local minima.

Question 9

Which of the following best describes the Mean Value Theorem?

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 10

A company’s profit $$P(x) (in $$dollars) from selling $$x$$ units is $$P(x) = -2x^2$ + 40x - 100. $$What is the number of units that maximizes profit?

Accepted Answer

$$x = 10$$

Question 11

Which of the following is the correct derivative of $$y = 	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 12

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 13

A population grows according to $$P(t) = 1000e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$ years?

Accepted Answer

$$30e^{0.3}$$ \approx 40.5$$

Question 14

Which of the following is the correct application of implicit differentiation for $$x^2$$ + $$y^2 = 25$?

Accepted Answer

$$2x + 2y \frac{dy}{dx} = 0$$

Question 15

Which of the following is the correct derivative of $$y = \arcsin(x)$$?

Accepted Answer

$$\frac{1}{\sqrt{1 - x^2$}}$$

Question 16

A tank is being filled with water such that the volume $$V(t) = 5t^2$ + 3t$$ liters. What is the rate at which water is entering the tank at $$t = 4$$ minutes?

Accepted Answer

$$43$$ liters/min

Question 17

Which of the following is the correct second derivative of $$f(x) = x^3$ - 6x^2$ + 9x$$?

Accepted Answer

$$6x - 12$$

Question 18

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 19

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

Accepted Answer

$$-2x\sin(x^2$)$$

Question 20

A bacteria culture grows according to $$N(t) = 500e$$^{0.2t}$$. What is the rate of growth at t = 5$ hours?

Accepted Answer

$$100e^{1}$$

Calculus I: Core Topics and Learning Objectives

Study Guide - Practice Questions

Study Guide - Flashcards