Question 1

Given the function $$f(x) = -4x^4$ - 3x^3$ - 36x^2$ + 8$$, find the $$x-$$coordinate of the absolute maximum on the interval $$[-6, 6].$$

Accepted Answer

$$x = -6$$

Question 2

A company’s total profit from the sale of $$x$$ hundred thousand tires is approximated by $$P(x) = -\frac{27}{2}x^2$ + 60x - 100. $$What value of $$x$$ maximizes the profit?

Accepted Answer

$$x = 2.22$$

Question 3

Given the demand function $$p = 80 - 10x$$, where $$p is $$the price per unit and $$x is $$the number of units sold, what is the elasticity of demand when $$x = 4$$?

Accepted Answer

$$E = 0.5$$

Question 4

Find $$\frac{dy}{dx} by $$implicit differentiation for $$3x^2y$$ + $$y^3 = 8$.

Accepted Answer

$$\frac{dy}{dx} = \frac{-6xy}{3x^2$ + 3y^2$}$$

Question 5

A hotel has 250 units. All rooms are occupied when the hotel charges $$\70$ per day per room. For every $$\$$2$$ increase in price, one fewer room is occupied. What price should the hotel charge per day to maximize daily revenue?

Accepted Answer

Question 6

Given $$y = x^2$ + 2x at $$the point $$(1, 3)$$, what is the equation of the tangent line?

Accepted Answer

$$y = 4x - 1$$

Question 7

If $$C(x) = 2x^2$ + 570$$ and $$R(x) = 51x$$, what is the rate of change of profit with respect to $$x$$?

Accepted Answer

$$P'(x) = 51 - 4x$$

Question 8

A streetlight is mounted on a pole 18 ft high. A man 6 ft tall walks away from the pole at a rate of 2 ft/sec. How fast is the tip of his shadow moving when he is 18 ft from the pole?

Accepted Answer

$$3 ft/sec$$

Question 9

Given $$y = \sqrt{x^2$ + 1}$$, find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 10

If $$x$$ and $$y$$ are functions of $$t$$ and $$z = x^2$y + 2x$$, $$\frac{dx}{dt} = 4$$, $$\frac{dy}{dt} = 1$$, $$x = 1$$, $$y = 1$$, what is $$\frac{dz}{dt}$$?

Accepted Answer

$$10$$

Question 11

A product sells for $$\20$ per unit. The company finds that revenue from sales is R(x) = 20x$$ - $$0.01x^2$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 500$?

Accepted Answer

$$5$$

Question 12

Find the equation of the tangent line to $$y = x^2$ - 2x at$$ $$x = 1.$$

Accepted Answer

$$y = 0x - 1$$

Question 13

If the price charged for a candy bar is $$p(x) = 50 - x^2$, then thousand candy bars will be sold in a city where p(x) = 32$. How many thousand candy bars must be sold to maximize revenue?

Accepted Answer

$$x = 5$$

Question 14

Given $$y = x^3$ - 2x^2$ + 4x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 15

A company’s cost and revenue for producing and selling $$x$$ units are $$C(x) = 2x^2$ + 570$$ and $$R(x) = 51x. $$What is the marginal profit when $$x = 10$$?

Accepted Answer

$$-29$$

Question 16

If $$y = x^2$ + 2x$$ and $$x = 1$$, what is the slope of the tangent line at this point?

Accepted Answer

$$4$$

Business Calculus: Extrema, Optimization, and Applications Study Guide

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