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 the answer with a proof or a counterexample.

367.

Every function has a derivative.

368.

A continuous function has a continuous derivative.

369.

A continuous function has a derivative.

370.

If a function is differentiable, it is continuous.

Use the limit definition of the derivative to exactly evaluate the derivative.

371.

f ( x ) = x + 4 f ( x ) = x + 4

372.

f ( x ) = 3 x f ( x ) = 3 x

Find the derivatives of the following functions.

373.

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

374.

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

375.

f ( x ) = e sin x f ( x ) = e sin x

376.

f ( x ) = ln ( x + 2 ) f ( x ) = ln ( x + 2 )

377.

f ( x ) = x 2 cos x + x tan ( x ) f ( x ) = x 2 cos x + x tan ( x )

378.

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

379.

f ( x ) = x 4 sin −1 ( x ) f ( x ) = x 4 sin −1 ( x )

380.

x 2 y = ( y + 2 ) + x y sin ( x ) x 2 y = ( y + 2 ) + x y sin ( x )

Find the following derivatives of various orders.

381.

First derivative of y=xln(x)cosxy=xln(x)cosx

382.

Third derivative of y=(3x+2)2y=(3x+2)2

383.

Second derivative of y=4x+x2sin(x)y=4x+x2sin(x)

Find the equation of the tangent line to the following equations at the specified point.

384.

y=cos−1(x)+xy=cos−1(x)+x at x=0x=0

385.

y=x+ex1xy=x+ex1x at x=1x=1

Draw the derivative for the following graphs.

386.
The function begins at (−3, 0.5) and decreases to a local minimum at (−2.3, −2). Then the function increases through (−1.5, 0) and slows its increase through (0, 2). It then slowly increases to a local maximum at (2.3, 6) before decreasing to (3, 3).
387.
The function decreases linearly from (−1, 4) to the origin, at which point it increases as x2, passing through (1, 1) and (2, 4).

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by w(t)=1.9+2.9cos(π6t),w(t)=1.9+2.9cos(π6t), where t is measured in hours after midnight, and the height is measured in feet.

388.

Find and graph the derivative. What is the physical meaning?

389.

Find w(3).w(3). What is the physical meaning of this value?

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

Hours after Midnight, August 26 Wind Speed (mph)
1 45
5 75
11 100
29 115
49 145
58 175
73 155
81 125
85 95
107 35
Table 3.9 Wind Speeds of Hurricane Katrina Source: http://news.nationalgeographic.com/news/2005/09/0914_050914_katrina_timeline.html.
390.

Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?

391.

Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

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.