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

Question 1

Given the function $$f(x) = x^2$, what is the equation of the tangent line at x = 1$?

Accepted Answer

$$y = 2x - 1$$

Question 2

A company’s cost and revenue functions are $$C(x) = 500 + 8x$$ and $$R(x) = 18x. $$What is the break-even quantity?

Accepted Answer

$$x = 50$$

Question 3

Which of the following best describes a removable discontinuity in a function’s graph?

Accepted Answer

The two-sided limit exists, but the function value is missing or differs from the limit.

Question 4

If $$p(x) = 60 - 2x is $$the demand function, what is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 60x - 2x^2$

Question 5

Which of the following is the correct definition of the derivative at a point $$a$$?

Accepted Answer

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

Question 6

For the rational function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what type of discontinuity occurs at $$x = 1$$?

Accepted Answer

Removable discontinuity

Question 7

If $$f(x) = (2x + 1)^5$, what is f$$'(x)$$?

Accepted Answer

$$10(2x + 1)^4$

Question 8

A business has a fixed cost of $$\500$ and a variable cost of $$\$$8$$ per unit. What is the total cost function $$C(x)$$?

Accepted Answer

$$C(x) = 500 + 8x$$

Question 9

Which rule should be used to differentiate $$f(x) = x^2$ \cdot e^x$$?

Accepted Answer

Product rule

Question 10

If $$f(x) = x^3$ - 2x^2$ + x$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2 - 4x + 1$

Question 11

Suppose $$C(x) = 500 + 8x$$ and $$R(x) = 18x. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 10x - 500$$

Question 12

Which of the following is the correct formula for the average rate of change of $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 13

If $$f(x) = \ln(x)$$, what is $$f'(x)$$ for $$x \u003e 0$$?

Accepted Answer

$$\frac{1}{x}$$

Question 14

Given $$f(x) = x^2$ + 3x + 2$$, what is the $$x-$$coordinate of the vertex?

Accepted Answer

$$x = -\frac{3}{2}$$

Question 15

Which of the following is the correct formula for the marginal cost if $$C(x) is $$the total cost function?

Accepted Answer

$$MC(x) = C'(x)$$

Question 16

If $$f(x) = \frac{1}{x}$$, what type of discontinuity occurs at $$x = 0$$?

Accepted Answer

Infinite discontinuity

Question 17

Which of the following is the correct formula for the derivative of $$f(x) = a^x$$ where $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

$$f'(x) = a^x \ln(a)$$

Question 18

If $$f(x) = x^2$ + 1$$ and $$g(x) = 3x - 4$$, what is the derivative of $$h(x) = f(x)g(x)$$?

Accepted Answer

$$h'(x) = 2x(3x - 4) + 3(x^2$ + 1)$$

Question 19

Which of the following is the correct formula for the change of base for logarithms?

Accepted Answer

$$\log_b(x) = \frac{\ln(x)}{\ln(b)}$$

Business Calculus Exam 1 Review: Functions, Limits, Derivatives, and Business Applications

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