Q: Which of the following is the correct application of the definition of the derivative for $$f(x) = x^2$ at x = 2$?

$$\lim_{h \to 0} \frac{(2+h)^2$ - 2^2$}{h}$$

Question 1

A function $$f(x) is $$defined as follows:

$$ f(x) = \begin{cases} x^2$ & x  1 \end{cases} $$

Which of the following statements about the continuity of $$f(x) at$$ $$x = 1 is $$correct?

Accepted Answer

The function is not continuous at $$x = 1$$ because $$\lim_{x 	o 1^-} f(x) 
eq \lim_{x 	o 1^+} f(x).$$

Question 2

Evaluate the following limit:

$$ \lim_{x 	o 0} \frac{\sin(5x)}{x} $$

Accepted Answer

$$5$$

Question 3

Which of the following is the derivative of $$f(x) = x^3$ - 5x^2$?

Accepted Answer

$$3x^2 - 10x$

Question 4

A medication's concentration in the bloodstream is modeled by $$C(t) = 8e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-4e^{-1}$$

Question 5

Find the value of the definite integral $$\int_{0}^{2} (3x^2$ - 2x) \, dx.$$

Accepted Answer

$$6$$

Question 6

Which of the following statements is true about the function $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$f(x) is $$discontinuous only at $$x = 0.$$

Question 7

Evaluate the following limit:

$$ \lim_{x 	o 3} \frac{x^2$ - 9}{x - 3} $$

Accepted Answer

$$6$$

Question 8

Find the derivative of $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

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

Question 9

A car's velocity is given by $$v(t) = 4t^3$ - 2t. $$What is the total distance traveled from $$t = 0 to$$ $$t = 2$$ seconds?

Accepted Answer

$$32$$

Question 10

Which of the following is the correct application of the definition of the derivative for $$f(x) = x^2$ at x = 2$?

Accepted Answer

$$\lim_{h 	o 0} \frac{(2+h)^2$ - 2^2$}{h}$$

Question 11

Evaluate the following indefinite integral:

$$ \int 5x^4$ \, dx $$

Accepted Answer

Question 12

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

Accepted Answer

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

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

$$20e^{0.2 	imes 5}$$

Question 14

Evaluate the following limit:

$$ \lim_{x 	o 0} \frac{e^x - 1}{x} $$

Accepted Answer

$$1$$

Question 15

Find the indefinite integral $$\int \frac{1}{x} \, dx.$$

Accepted Answer

$$\ln|x| + C$$

Question 16

Which of the following is the correct derivative of $$f(x) = \frac{1}{x^2$}$$?

Accepted Answer

$$-2x^{-3}$$

Question 17

A tank is being filled with water at a rate given by $$r(t) = 3t^2$ liters per minute. How much water is added between t = 0$ and t = 4$ minutes?

Accepted Answer

$$64$$ liters

Question 18

Find the derivative of $$f(x) = \sqrt{x^2$ + 4}.$$

Accepted Answer

$$\frac{x}{\sqrt{x^2$ + 4}}$$

Question 19

Evaluate the following definite integral:

$$ \int_{0}^{\pi} \sin(x) \, dx $$

Accepted Answer

$$0$$

Calculus I Final Exam Review: Limits, Derivatives, and Integrals

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