Q: A hospital is monitoring the spread of a virus. The number of infected patients $$N(t) at $$time $$t (in $$days) is modeled by $$N(t) = 200e$$^{0.05t}$$. What is the rate of change of infected patients after 10 days?

$$N'(10) = 200 \times 0.05 \times e$$^{0.5}$$

Q: A medication is administered such that its concentration in the bloodstream is $$C(t) = 5e$$^{-0.2t}$$ mg/L, where t$ is in hours. What is the instantaneous rate of change of concentration at t = 3$ hours?

$$C'(3) = -0.2 \times 5e$$^{-0.6}$$

Q: Let $$f(x) = \frac{1}{x}. $$Which of the following statements is TRUE?

$$\lim_{x \to 0^+} f(x) = \infty$$

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

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

Q: Let $$f(x) = \frac{1}{x^2$}. $$Which of the following statements is TRUE?

$$\lim_{x \to 0^+} f(x) = \infty$$

Question 1

A hospital is monitoring the spread of a virus. The number of infected patients $$N(t) at $$time $$t (in $$days) is modeled by $$N(t) = 200e$$^{0.05t}$$. What is the rate of change of infected patients after 10 days?

Accepted Answer

$$N'(10) = 200 	imes 0.05 	imes e$$^{0.5}$$

Question 2

Let $$f(x) = \sqrt{x^2$ + 1}. $$What is the equation of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = \frac{2}{\sqrt{5}}(x-2) + \sqrt{5}$$

Question 3

A ball is thrown upward from the ground with a velocity of $$20 m/s. $$Its height after $$t$$ seconds is $$h(t) = 20t - 4.9t^2$. At what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04 s$$

Question 4

Which of the following limits does NOT exist?

Accepted Answer

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

Question 5

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

Accepted Answer

$$4$$

Question 6

A medication is administered such that its concentration in the bloodstream is $$C(t) = 5e$$^{-0.2t}$$ mg/L, where t$ is in hours. What is the instantaneous rate of change of concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -0.2 	imes 5e$$^{-0.6}$$

Question 7

Suppose $$g(x) = \sin(2x). $$What is $$g'(\frac{\pi}{4})$$?

Accepted Answer

$$2\cos(\frac{\pi}{2})$$

Question 8

Let $$f(x) = \frac{1}{x}. $$Which of the following statements is TRUE?

Accepted Answer

$$\lim_{x 	o 0^+} f(x) = \infty$$

Question 9

A chemical reaction's rate is given by $$R(t) = 3t^2$ - 2t + 1. $$What is the average rate of change of $$R$$ from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$R_{avg} = 11$$

Question 10

Let $$f(x) = x^3$ - 3x^2$ + 2x. $$For which value of $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 2$$

Question 11

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

Accepted Answer

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

Question 12

A population of bacteria doubles every 3 hours. If the initial population is 100, what is the population after 9 hours?

Accepted Answer

$$800$$

Question 13

Let $$f(x) = \frac{2x+1}{x-1}. $$What is the vertical asymptote of $$f(x)$$?

Accepted Answer

$$x = 1$$

Question 14

Suppose $$f(x) is $$continuous on $$[1, 3]$$ and $$f(1) = -2$$, $$f(3) = 4. $$Which theorem guarantees that $$f(x) = 0$$ for some $$x in$$ $$(1, 3)$$?

Accepted Answer

Intermediate Value Theorem

Question 15

Let $$f(x) = \sin(x). $$What is $$\lim_{x 	o 0} \frac{\sin(x)}{x}$$?

Accepted Answer

$$1$$

Question 16

A car's position is given by $$s(t) = t^3$ - 6t^2$ + 9t (in $$meters, $$t in $$seconds). What is the car's acceleration at $$t = 2$$?

Accepted Answer

$$s''(2) = 0$$

Question 17

Let $$f(x) = \frac{1}{x^2$}. $$Which of the following statements is TRUE?

Accepted Answer

$$\lim_{x 	o 0^+} f(x) = \infty$$

Question 18

Suppose $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$. What is (f$$ \circ g)(4)$$?

Accepted Answer

$$4$$

Question 19

Let $$f(x) = \frac{x^2$ - 9}{x - 3}$$ for $$x 
eq 3. $$What is $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Calculus I: Comprehensive Study Guide for Midterm Exam Preparation

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