Question 1

Which of the following statements best describes the meaning of the limit $$\displaystyle \lim_{x 	o 3} f(x) = 7$$?

Accepted Answer

As $$x$$ approaches $$3$$, $$f(x)$$ becomes arbitrarily close to $$7.$$

Question 2

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

Accepted Answer

$$5$$

Question 3

A hospital monitors the temperature of a patient over time, modeled by $$T(t) = 98 + 2e$$^{-0.5t}$$, where t$ is in hours. What is the instantaneous rate of change of the patient's temperature at t = 2$ hours?

Accepted Answer

$$T'(2) = -e$$^{-1}$$

Question 4

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

Accepted Answer

$$f(x) = \frac{x^2$ - 4}{x - 2}$$

Question 5

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

Accepted Answer

$$\frac{3}{5}$$

Question 6

A chemical reaction's rate is modeled by $$R(t) = 4t^3$ - 2t + 7. $$What is the instantaneous rate of change of the reaction at $$t = 1$$?

Accepted Answer

$$R'(1) = 10$$

Question 7

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

Accepted Answer

$$\frac{1}{\sin(x)} \cos(x)$$

Question 8

A company wants to minimize the cost $$C(x) = x^2$ - 8x + 20$$ for producing $$x$$ units. What is the minimum cost and at how many units does it occur?

Accepted Answer

Minimum cost is $$4 at$$ $$x = 4$$ units.

Question 9

Which of the following is a correct application of the Mean Value Theorem for $$f(x) = x^2$ on $$[1, 5]$$?

Accepted Answer

There exists $$c in$$ $$(1, 5)$$ such that $$f'(c) = \frac{f(5) - f(1)}{5 - 1}.$$

Question 10

Evaluate $$\displaystyle \$$int_0$$^2 (3x^2$ + 4) \, dx.$$

Accepted Answer

$$16$$

Question 11

A tank is being filled with water at a rate given by $$r(t) = 5 + 2t$$ liters per minute. How much water enters the tank from $$t = 0 to$$ $$t = 3$$ minutes?

Accepted Answer

$$24$$ liters

Question 12

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

Accepted Answer

$$1$$

Question 13

Which of the following is the correct antiderivative of $$f(x) = 6x^2$?

Accepted Answer

$$2x^3 + C$

Question 14

A ladder 10 ft long is leaning against a wall. If the bottom slides away from the wall at $$2 ft/sec$$, how fast is the top sliding down when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-1.5 ft/sec$$

Question 15

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

Accepted Answer

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

Question 16

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b). $$Which of the following must be true?

Accepted Answer

$$f(x) is $$increasing on $$(a, b).$$

Question 17

Evaluate $$\displaystyle \int x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

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

Question 18

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It connects differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 19

A medication is administered at a rate $$r(t) = 10e$$^{-0.2t}$$ mg/hour. What is the total amount administered from t = 0$ to t = 5$ hours?

Accepted Answer

$$25(1 - e$$^{-1}$$) mg$$

Calculus I: Limits, Derivatives, and Introduction to Integration – Study Guide

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