Q: A biologist models bacteria growth with $$N(t) = 100e$$^{0.05t}$$. What is the rate of growth at t = 10$ hours?

$$5 \times 100e$$^{0.5}$$ \approx 824$$

Q: Which of the following limits is an indeterminate form of type $$\frac{0}{0}$$?

$$\lim_{x \to 0} \frac{\sin x}{x}$$

Question 1

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and differentiable on $$(1, 4)$$ with $$f(1) = 2$$ and $$f(4) = 8. $$According to the Mean Value Theorem, which of the following must be true?

Accepted Answer

$$f'(c) = 3$$ for some $$c in$$ $$(1, 4)$$

Question 2

Let $$f(x) = x^3$ - 3x^2$ + 2. On $$which interval is $$f(x)$$ increasing?

Accepted Answer

$$(2, \infty)$$

Question 3

If $$f(x) is an $$invertible function and $$f(2) = 5$$, what is $$f^{-1}(5$$)$$?

Accepted Answer

$$2$$

Question 4

Which of the following is the derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$2e^{2x}$$

Question 5

A biologist models bacteria growth with $$N(t) = 100e$$^{0.05t}$$. What is the rate of growth at t = 10$ hours?

Accepted Answer

$$5 	imes 100e$$^{0.5}$$ \approx 824$$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

$$\cosh(x)$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{\sin(5x)}{x}$$ using L'Hospital's Rule.

Accepted Answer

$$5$$

Question 10

A company’s profit function is $$P(x) = -2x^2$ + 8x + 10. $$What is the maximum profit, and at what value of $$x$$ does it occur?

Accepted Answer

Maximum profit is $$18 at$$ $$x = 2$$

Question 11

If $$f'(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing

Question 12

Which of the following best describes the concavity of $$f(x) if$$ $$f''(x) \u003c 0$$ for all $$x in an $$interval?

Accepted Answer

$$f(x) is $$concave down

Question 13

Given $$f(x) = x^4$ - 4x^2$, which of the following points is a local minimum?

Accepted Answer

$$(\sqrt{2}, -4)$$

Question 14

Which of the following is the correct derivative of $$f(x) = \log_{10}(x)$$?

Accepted Answer

$$\frac{1}{x \ln(10)}$$

Question 15

Suppose $$y = f$$^{-1}$$(x)$$ and $$f(3) = 7$$, $$f'(3) = 4. $$What is $$(f^{-1}$$)'(7)$$?

Accepted Answer

$$\frac{1}{4}$$

Question 16

Which of the following limits is an indeterminate form of type $$\frac{0}{0}$$?

Accepted Answer

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

Question 17

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

Accepted Answer

$$\sinh(x)$$

Question 18

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = 50e$$^{-0.2t}$$. What is the rate of change of concentration at t = 5$ hours?

Accepted Answer

$$-10e^{-1}$$

Question 19

Which of the following statements about curve sketching is correct?

Accepted Answer

Points where $$f'(x) = 0$$ and $$f''(x) \u003e 0$$ are local minima

Key Concepts in Differential Calculus: Theorems, Tests, and Applications

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