Question 1

Suppose the curve is defined implicitly by $$x^2$$ + $$y^2 = 25$. What is the slope of the tangent line at the point (3$$, 4)$$?

Accepted Answer

$$-\frac{3}{4}$$

Question 2

Given $$f(x) = x^x$$, use logarithmic differentiation to find $$f'(x). $$Which of the following is correct?

Accepted Answer

$$x^x \cdot (1 + \ln(x))$$

Question 3

A company’s demand function is $$p = 100 - 2q. $$What is the price elasticity of demand at $$q = 20$$?

Accepted Answer

$$-1$$

Question 4

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 1 on $$the interval $$[0, 5]$$, what is the absolute maximum value?

Accepted Answer

$$16$$

Question 5

Find the linearization $$L(x) of$$ $$f(x) = \sqrt{x} at$$ $$x = 4. $$Use $$L(x) to $$approximate $$\sqrt{4.1}.$$

Accepted Answer

$$2.05$$

Question 6

If the daily revenue function is $$R(q) = 50q - q^2$, what is the marginal revenue when q = 10$?

Accepted Answer

$$30$$

Question 7

A company has total revenue $$TR(q) = 100q - 2q^2$ and total cost TC(q) = 20q + 10$. What is the number of units q$ that maximizes profit?

Accepted Answer

$$20$$

Question 8

Given the demand function $$p = 200 - 4q$$, what is the price that maximizes revenue?

Accepted Answer

$$100$$

Question 9

For the function $$f(x) = x^2$ e^x$$, use logarithmic differentiation to find $$f'(x). $$Which is correct?

Accepted Answer

$$x^2 e^x$$ (\frac{2}{x} + 1)$$

Question 10

A pharmaceutical company models the cost of producing $$q$$ units as $$C(q) = 5q^2$ + 20q + 100. $$What is the marginal cost when $$q = 10$$?

Accepted Answer

$$140$$

Question 11

Suppose the curve $$xy = 12$$ passes through the point $$(3, 4). $$What is the equation of the tangent line at this point?

Accepted Answer

$$y = -\frac{4}{3}x + 8$$

Question 12

A company’s revenue function is $$R(q) = 60q - 0.5q^2$. What is the maximum daily revenue?

Accepted Answer

$$1800$$

Question 13

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

Accepted Answer

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

Question 14

A company’s profit function is $$P(q) = 80q - 3q^2$ - 20q - 100. $$What is the maximum profit?

Accepted Answer

$$600$$

Question 15

If $$f(x) = x^2$ + 2x$$, what is the linearization at $$x = 1$$?

Accepted Answer

$$L(x) = 3 + 4(x - 1)$$

Question 16

A company’s demand function is $$p = 120 - 3q. At$$ $$q = 30$$, what type of elasticity does the demand have?

Accepted Answer

Unit elastic

Question 17

Given $$f(x) = x^2$ + 4x + 5$$, what is the absolute minimum value on $$[-2, 2]$$?

Accepted Answer

$$5$$

Question 18

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

Accepted Answer

$$10$$

Question 19

If the demand function is $$p = 80 - q$$, what is the price elasticity of demand at $$q = 40$$?

Accepted Answer

$$-1$$

Question 20

Given $$f(x) = x^3$, what is the linearization at x = 2$?

Accepted Answer

$$L(x) = 8 + 12(x - 2)$$

Business Calculus: Differentiation, Applications, and Optimization Study Guide

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