Q: Which of the following is the correct application of L'Hôpital's Rule to $$\lim_{x \to 0} \frac{e^x - 1}{x}$$?

$$\lim_{x \to 0} \frac{e^x}{1} = 1$$

Q: A ball is thrown upward and its height at time $$t is$$ $$h(t) = 10t - 4.9t^2$. What is the velocity of the ball at t = 2$ seconds?

$$10 - 9.8 \times 2 = -9.6 m/s$$

Question 1

A car's velocity at time $$t is $$given by $$v(t) = 3t^2$ + 2t. $$What is the average velocity of the car between $$t = 1$$ and $$t = 4$$ seconds?

Accepted Answer

$$\frac{1}{4-1} \int_{1}^{4} (3t^2$ + 2t) \, dt$$

Question 2

Which of the following statements about the continuity of a function $$f(x) is $$TRUE?

Accepted Answer

A function $$f(x) is $$continuous at $$x = a if$$ $$\lim_{x 	o a} f(x)$$ exists, $$f(a)$$ exists, and $$\lim_{x 	o a} f(x) = f(a).$$

Question 3

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

Accepted Answer

$$5$$

Question 4

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = e$$^{-0.5t}$$. What is the instantaneous rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-0.5 e$$^{-1}$$

Question 5

Which theorem guarantees that a continuous function $$f(x) on$$ $$[a, b]$$ takes on every value between $$f(a)$$ and $$f(b)$$?

Accepted Answer

Intermediate Value Theorem

Question 6

Find the derivative of $$f(x) = x^2$ \sin(x)$$ using the product rule.

Accepted Answer

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

Question 7

A population of bacteria grows according to $$P(t) = 1000 e$$^{0.2t}$$. What is the rate of change of the population at t = 5$ hours?

Accepted Answer

$$200 e$$^{0.2 	imes 5}$$

Question 8

Which of the following is the correct application of the chain rule to $$f(x) = \cos(3x^2$)$$?

Accepted Answer

$$-\sin(3x^2$) \cdot 6x$$

Question 9

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

Accepted Answer

$$\int_{0}^{3} 4t \, dt$$

Question 10

Which of the following is the correct statement of the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If $$F(x) = \int_{a}^{x} f(t) \, dt$$, then $$F'(x) = f(x)$$

Question 11

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 12

A hospital monitors the temperature of a patient using $$T(t) = 98.6 + 0.5 \sin(\pi t)$$, where $$t is in $$hours. What is the rate of change of temperature at $$t = 2$$ hours?

Accepted Answer

$$0.5 \pi \cos(2\pi)$$

Question 13

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 14

A company wants to maximize profit given by $$P(x) = -2x^2$ + 12x - 20. $$What value of $$x$$ gives the maximum profit?

Accepted Answer

$$x = 3$$

Question 15

Evaluate the definite integral $$\int_{0}^{\pi} \sin(x) \, dx.$$

Accepted Answer

$$0$$

Question 16

Which of the following is the correct application of L'Hôpital's Rule to $$\lim_{x 	o 0} \frac{e^x - 1}{x}$$?

Accepted Answer

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

Question 17

Find the area between the curves $$y = x^2$ and y = x$ from x = 0$ to x = 1$.

Accepted Answer

$$\int_{0}^{1} (x - x^2$) \, dx$$

Question 18

A ball is thrown upward and its height at time $$t is$$ $$h(t) = 10t - 4.9t^2$. What is the velocity of the ball at t = 2$ seconds?

Accepted Answer

$$10 - 9.8 	imes 2 = -9.6 m/s$$

Question 19

Which of the following is the correct linear approximation of $$f(x) = \sqrt{x} at$$ $$x = 4$$?

Accepted Answer

$$f(x) \approx 2 + \frac{1}{4}(x - 4)$$

Calculus I (MATH-2413) Syllabus and Core Concepts Study Guide

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