Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Calculus Volume 1

Review Exercises

Calculus Volume 1Review Exercises

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

310.

A function is always one-to-one.

311.

fg=gf,fg=gf, assuming f and g are functions.

312.

A relation that passes the horizontal and vertical line tests is a one-to-one function.

313.

A relation passing the horizontal line test is a function.

For the following problems, state the domain and range of the given functions:

f=x2+2x3,g=ln(x5),h=1x+4f=x2+2x3,g=ln(x5),h=1x+4

314.

h

315.

g

316.

hfhf

317.

gfgf

Find the degree, y-intercept, and zeros for the following polynomial functions.

318.

f(x)=2x2+9x5f(x)=2x2+9x5

319.

f(x)=x3+2x22xf(x)=x3+2x22x

Simplify the following trigonometric expressions.

320.

tan 2 x sec 2 x + cos 2 x tan 2 x sec 2 x + cos 2 x

321.

cos 2 x - sin 2 x cos 2 x - sin 2 x

Solve the following trigonometric equations on the interval x=[−2π,2π]x=[−2π,2π] exactly.

322.

6cos2x3=06cos2x3=0

323.

sec 2 x 2 sec x + 1 = 0 sec 2 x 2 sec x + 1 = 0

Solve the following logarithmic equations.

324.

5 x = 16 5 x = 16

325.

log 2 ( x + 4 ) = 3 log 2 ( x + 4 ) = 3

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse f−1(x)f−1(x) of the function. Justify your answer.

326.

f(x)=x2+2x+1f(x)=x2+2x+1

327.

f ( x ) = 1 x f ( x ) = 1 x

For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.

328.

f ( x ) = 9 x f ( x ) = 9 x

329.

f ( x ) = x 2 + 3 x + 4 f ( x ) = x 2 + 3 x + 4

330.

A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.

331.

a. Find the equation C=f(x)C=f(x) that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

332.

a. Find the inverse function x=f−1(C)x=f−1(C) and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend.

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

333.

The population can be modeled by P(t)=82.567.5cos[(π/6)t],P(t)=82.567.5cos[(π/6)t], where tt is time in months (t=0(t=0 represents January 1) and PP is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

334.

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as P(t)=82.567.5cos[(π/6)t]+t,P(t)=82.567.5cos[(π/6)t]+t, where tt is time in months (t=0t=0 represents January 1) and PP is population (in thousands). When is the first time the population reaches 200,000?

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation y=ert,y=ert, where yy is the ratio of radiocarbon still present in the material, tt is the number of years passed, and r=−0.0001210r=−0.0001210 is the decay rate of radiocarbon.

335.

If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

336.

Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction
Citation information

© Jul 25, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.