Q: Suppose the population of a bacteria culture grows according to $$P(t) = 100e$$^{0.03t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$ hours?

$$P'(10) = 3 \times 100e$$^{0.3}$$

Q: A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate at which the concentration is changing at t = 4$ hours?

$$C'(4) = -0.2 \times 5e$$^{-0.8}$$

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

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

Question 1

Which of the following statements best describes the limit of a function $$f(x) as$$ $$x$$ approaches $$a$$?

Accepted Answer

The value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a$$, regardless of $$f(a).$$

Question 2

Suppose the population of a bacteria culture grows according to $$P(t) = 100e$$^{0.03t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$ hours?

Accepted Answer

$$P'(10) = 3 	imes 100e$$^{0.3}$$

Question 3

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

Accepted Answer

$$y' = \cos(3x^2$ + 2x) \cdot (6x + 2)$$

Question 4

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate at which the concentration is changing at t = 4$ hours?

Accepted Answer

$$C'(4) = -0.2 	imes 5e$$^{-0.8}$$

Question 5

Evaluate the following limit using L'Hospital's Rule: $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

1

Question 6

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

Accepted Answer

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

Question 7

A ball is thrown upward with a velocity $$v(t) = 20 - 9.8t (in m/s). At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$ seconds

Question 8

Which of the following statements about inverse functions is TRUE?

Accepted Answer

If $$f is $$invertible, then $$f^{-1}(f(x)) = x$ for all x$ in the domain of f$.

Question 9

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find all critical points and determine their nature (local max, min, or neither).

Accepted Answer

Critical points at $$x = 0 ($$max), $$x = 2 ($$min)

Question 10

A company’s profit $$P(x) (in $$thousands of dollars) from selling $$x$$ units is given by $$P(x) = -2x^2$ + 40x - 100. $$How many units should be sold to maximize profit?

Accepted Answer

$$x = 10$$

Question 11

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 12

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$ + 2x) dx.$$

Accepted Answer

16

Question 13

Which of the following best describes the meaning of the derivative in a real-world context?

Accepted Answer

The derivative gives the instantaneous rate of change of a quantity.

Question 14

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees that there exists $$c \in (a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 15

A tank is being filled with water at a rate given by $$R(t) = 4t$$ liters per minute. How much water is added to the tank between $$t = 0$$ and $$t = 5$$ minutes?

Accepted Answer

50 liters

Question 16

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

Accepted Answer

$$y' = \frac{4x}{2x^2$ + 1}$$

Question 17

If $$f(x) = e$$^{x^2}$$, what is f$$'(x)$$?

Accepted Answer

$$2xe^{x^2}$$

Question 18

A car’s position is given by $$s(t) = t^3$ - 6t^2$ + 9t (in $$meters). At what time(s) is the car at rest?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 19

Which of the following is the correct antiderivative of $$f(x) = \cos x$$?

Accepted Answer

$$\sin x + C$$

Question 20

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and $$f(1) = 2$$, $$f(4) = 8. $$Which theorem guarantees that there is some $$c \in (1, 4)$$ such that $$f(c) = 5$$?

Accepted Answer

Intermediate Value Theorem

Calculus I Syllabus and Study Guide: Limits, Derivatives, and Applications

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