$$\lim_{x \to 0} \dfrac{1}{x}$$

$$f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$$

Question 1

Given the function $$f(x) = 2x^2$ - 3x + 5$$, what is the instantaneous rate of change of $$f at$$ $$x = 1$$?

Accepted Answer

$$-1$$

Question 2

Which of the following best describes the break-even point in a business context?

Accepted Answer

The point where total revenue equals total cost.

Question 3

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 4

A company’s cost function is $$C(q) = 100 + 5q. $$What is the marginal cost when $$q = 10$$?

Accepted Answer

$$5$$

Question 5

Evaluate $$\lim_{x 	o 0} \dfrac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 6

Which of the following statements about the function $$f(x) = e$$^{2x}$$ is true?

Accepted Answer

$$f(x) is $$always positive for all real $$x.$$

Question 7

Suppose the demand and supply for a product are given by $$D(p) = 100 - 2p$$ and $$S(p) = 4p$$, respectively. What is the market equilibrium price?

Accepted Answer

$$p = 20$$

Question 8

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at what $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 9

Which of the following is the correct derivative of $$f(x) = 5e$$^{3x}$$?

Accepted Answer

$$15e^{3x}$$

Question 10

A population of bacteria grows according to $$P(t) = 200e$$^{0.05t}$$. What is the population after 20 hours?

Accepted Answer

$$200e^{1}$$

Question 11

Which of the following limits does NOT exist?

Accepted Answer

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

Question 12

If $$f(x) = \sqrt{x}$$, what is $$f'(4)$$?

Accepted Answer

$$\dfrac{1}{4}$$

Question 13

A company’s revenue function is $$R(q) = 50q - q^2$. What is the maximum revenue, and at what quantity q$ does it occur?

Accepted Answer

Maximum revenue is $$625 at$$ $$q = 25.$$

Question 14

Which of the following is the correct application of the sum rule for derivatives?

Accepted Answer

$$\dfrac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

Question 15

If $$f(x) = \ln(x)$$ and $$g(x) = x^2$, what is $$\dfrac{d}{dx}[f(g(x))]$$?

Accepted Answer

$$\dfrac{2}{x}$$

Question 16

A medication decays in the bloodstream according to $$Q(t) = 100e$$^{-0.1t}$$. What is the amount remaining after 10 hours?

Accepted Answer

$$100e^{-1}$$

Question 17

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

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \dfrac{f(x+h) - f(x)}{h}$$

Question 18

If $$f(x) = x^2$ and g(x) = 3x$, what is $$\dfrac{d}{dx}[f(x) + g(x)]$$?

Accepted Answer

$$2x + 3$$

Question 19

Which of the following is the correct relationship between exponential and logarithmic functions?

Accepted Answer

$$\log_b(a) = c if $$and only if $$b^c = a$$

Question 20

If $$f(x) = 4x^3$ - 2x$$, what is the slope of the tangent line to the graph at $$x = 2$$?

Accepted Answer

$$44$$

Calculus I Exam 1 Study Guide: Functions, Limits, Derivatives, Exponential & Logarithmic Functions

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