$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

A limit is the value that a function approaches as the input approaches a certain value.

Question 2

Evaluate the following limit: $$\lim_{x 	o 2} (3x^2$ - 4x + 1)$$

Accepted Answer

$$5$$

Question 3

Which of the following statements about the derivative is correct?

Accepted Answer

The derivative of a function at a point gives the slope of the tangent line at that point.

Question 4

Find the derivative of $$f(x) = x^3$ - 2x + 5.$$

Accepted Answer

$$f'(x) = 3x^2$ - 2$$

Question 5

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

Accepted Answer

$$26 m/s$$

Question 6

Which rule is used to find the derivative of the product of two functions?

Accepted Answer

Product rule

Question 7

If $$f(x) = x^2$ and g(x$$) = \sin(x)$$, what is the derivative of h(x) = f(x)g(x$$)$$ using the product rule?

Accepted Answer

$$h'(x) = 2x\sin(x) + x^2$\cos(x)$$

Question 8

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

Accepted Answer

$$f'(x) = 3\cos(3x)$$

Question 9

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

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 10

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 11

Given $$f(x) = \frac{x^2$ - 4}{x - 2}$$, find $$\lim_{x 	o 2} f(x).$$

Accepted Answer

$$2$$

Question 12

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 13

A population of bacteria grows according to $$P(t) = 100e$$^{0.2t}$$. What is the instantaneous rate of growth at t = 5$ hours?

Accepted Answer

$$P'(5) = 20e$$^{1}$$

Question 14

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 15

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 16

A chemical reaction's rate is given by $$R(t) = 5t^3$ - 2t. $$What is the acceleration of the reaction rate at $$t = 2$$?

Accepted Answer

$$R''(2) = 60$$

Question 17

Which of the following is the correct derivative of $$f(x) = \frac{1}{x}$$?

Accepted Answer

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

Question 18

A ball is thrown upward and its height is given by $$h(t) = 20t - 4.9t^2$. What is the velocity of the ball at t = 2$ seconds?

Accepted Answer

$$h'(2) = 10.4 m/s$$

Question 19

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$}{x+1}$$?

Accepted Answer

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

Calculus 1A (Math 1776) Syllabus and Study Guide

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