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?

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 $z = x^2y + 2x$, $\frac{dx}{dt} = 4$, $\frac{dy}{dt} = 1$, $x = 1$, $y = 1$, 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

Study Guide - Practice Questions

Study Guide - Flashcards