Question 1

Evaluate the following definite integral, which models the total accumulated profit over a time period: $\int_0^2 (6e^{3x})\,dx$.

Question 2

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

Accepted Answer

$2x e^{3x} + 3x^2 e^{3x}$

Question 3

Find the critical values, if any, for $f(x) = xe^{-x^2}$.

Accepted Answer

$x = 0, x = \frac{1}{\sqrt{2}}$

Question 4

A company’s marginal profit function is $P'(x) = 5x^2 - 10x + 6$. If $P(2) = 20$, find the profit function $P(x)$.

Accepted Answer

$P(x) = \frac{5}{3}x^3 - 5x^2 + 6x + 4$

Question 5

The height $h$ (in meters) of a plant grows so that the rate of change is given by $\frac{dh}{dt} = 3t^2 + 2t$, where $t$ is in weeks. If $h(0) = 5$, what is $h(2)$?

Accepted Answer

$h(2) = 21$

Question 6

Evaluate $\int (5x^3 + 8x^2 - x)\,dx$.

Accepted Answer

$\frac{5}{4}x^4 + \frac{8}{3}x^3 - \frac{1}{2}x^2 + C$

Question 7

A company’s marginal cost function is $C'(x) = 160 - e^{-0.2x}$, where $x$ is the number of items produced. If the fixed cost is $500$, find the total cost function $C(x)$.

Accepted Answer

$C(x) = 160x + 5e^{-0.2x} + 500$

Question 8

Find the area bounded by the curves $y = 5 - x$ and $y = x^2$.

Accepted Answer

$\int_{-1}^{2} [(5-x) - x^2] dx$

Question 9

Given $f(x) = x^3 - 6x^2 + 9x + 1$, find all local maxima and minima.

Accepted Answer

Local max at $x = 1$, local min at $x = 3$

Question 10

A retailer sells calculators for $p$ dollars each. The demand function is $p = 50 - 2x$, where $x$ is the number of calculators sold. What price maximizes revenue?

Accepted Answer

$p = 25$

Question 11

Given $f''(x) = -6x + 12$, determine where the original function $f(x)$ is concave up.

Accepted Answer

$x > 2$

Question 12

A company’s marginal revenue function is $R'(x) = 100 - 4x$. If fixed revenue is $0$, what is the total revenue function $R(x)$?

Accepted Answer

$R(x) = 100x - 2x^2$

Question 13

Find the absolute maximum and minimum values of $f(x) = x^3 - 3x^2 + 1$ on the interval $[0, 3]$.

Accepted Answer

Max at $x=3$, min at $x=2$

Question 14

If $y = \ln(x^2 + 5y^2)$, find $\frac{dy}{dx}$ using implicit differentiation.

Question 15

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. How many units should be produced to maximize profit?

Accepted Answer

$x = 10$

Question 16

Evaluate $\lim_{x 	o 0} \frac{e^{2x} - 1}{x}$.

Accepted Answer

$2$

Question 17

Given $f(x) = x^2 e^{-x}$, find all critical points.

Accepted Answer

$x = 0, x = 2$

Question 18

A model for advertising is given by $M(q) = 600 \ln(q+500)$, where $q$ is the number of units sold. What does $M'(q)$ represent?

Accepted Answer

The marginal effect of selling one more unit on advertising impact

Question 19

Given $f(x) = 40x^{0.5}$, where $f(x)$ is in millions of dollars of finished product, and $x$ is in billions of dollars spent on labor, find the marginal productivity $f'(x)$ at $x = 10$.

Accepted Answer

$f'(10) = 2$

Business Calculus: First Exam Review Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards