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

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

Q: Which of the following limits requires the use of L'Hospital's Rule to evaluate?

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

Question 1

A hospital monitors the temperature of a patient over time. The temperature function is given by $$T(t) = 98.6 + 0.5 \sin(\pi t)$$, where $$t is in $$hours. What is the instantaneous rate of change of the patient's temperature at $$t = 2$$ hours?

Accepted Answer

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

Question 2

Which of the following statements about the limit $$\lim_{x 	o 3} (2x^2$ - 5x + 1) is $$correct?

Accepted Answer

The limit exists and equals $$2(3)^2 - 5(3) + 1 = 7$

Question 3

A chemical reaction rate is modeled by $$R(t) = \frac{5t}{t^2$ + 1}. $$What is $$\lim_{t 	o \infty} R(t)$$?

Accepted Answer

$$0$$

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

$$38$$

Question 6

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

Accepted Answer

$$5$$

Question 7

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

Accepted Answer

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

Question 8

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.2t}$$. What is the instantaneous rate of change of concentration at t=0$?

Accepted Answer

$$-2$$

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

$$30e^{0.3}$$

Question 11

Which of the following limits requires the use of L'Hospital's Rule to evaluate?

Accepted Answer

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

Question 12

A ball is dropped from a height and its position is given by $$s(t) = 100 - 4.9t^2$. What is the velocity after 3 seconds?

Accepted Answer

$$-29.4$$

Question 13

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$(fg)' = f'g + fg'$$

Question 14

Find the derivative of $$f(x) = x^2$ e^x.$$

Accepted Answer

$$e^x(x^2$ + 2x)$$

Question 15

A biologist models the growth of bacteria with $$N(t) = 500 \ln(t+1). $$What is the rate of change of the bacteria population at $$t=4$$?

Accepted Answer

$$100$$

Question 16

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

Accepted Answer

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

Question 17

A medication is administered and its concentration is modeled by $$C(t) = 8e$$^{-0.5t}$$. After how many hours is the rate of change of concentration -2$?

Accepted Answer

$$t = \ln(2)$$

Question 18

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

Accepted Answer

$$6x \cos(3x^2$)$$

Question 19

A company’s profit is modeled by $$P(x) = x^3$ - 6x^2$ + 9x + 15. At $$what value of $$x$$ does the profit have a critical point?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Calculus Homework Assignment List: Math LPC 1 Sec D08 (21409) Fall 2025

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