Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Continuing to Payment will take you to apayment page

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews March 4, 2024February 26, 2024

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes PLUS
Annual Plan - Group Discount

Qty: 00

SubtotalUS $0,000.00

Discount (00% off)
-US $000.00

TaxUS $XX.XX

DUE NOWUS $1,049.58

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

This equals zero when x = 0 or x = 4.
The direction of f (x) based on the sign of the derivative is depicted below:

f''(x)

=

=

=

This never equals zero. The concavity of the graph is depicted below:
To find the vertical asymptotes, take limits near the suspected asymptote:

= - ∞

= ∞

So, x = 2 is a vertical asymptote of the graph
To find the horizontal asymptotes, take limits at infinity:

= + ∞

= - ∞

Thus, f grows without bound, and there are no horizontal asymptotes
Now, find the exact coordinates of the intercepts and critical points:
y-intercept: (0, 0)x-intercept: (0, 0)
Critical Point: (0, 0) and (4, 8)
Combining the information from the first derivative work and the second derivative work
generates this single chart:
Based on this information, the graph may now be sketched:

Problem :
Give a possible equation of the function graphed below:

To find an appropriate equation, begin by pulling out relevant features of the graph.
f has vertical asymptotes at x = 3 and x = - 3. Vertical asymptotes can occur where
the denominator of a rational function is equal to zero. Thus, the function in question
should look something like:

f (x) =

This is the same as

f (x) =

f has a horizontal asymptote at y = 2, which means that as x approaches infinity (or
negative infinity), the function approaches 2. Thus, the function must look something
like:

f (x) =

Finally, f (0) = - 3, so the equation must be modified to:

f (x) =

Problem :
Give a possible equation of the function graphed below:

As in the last problem, start with the vertical asymptotes:

f (x) =

f (x) =

Because of the horizontal asymptotes,

f (x) =

f (0) = 0, but this does not modify the equation of the function.

Problem :
Below is a sketch of the graph of f'(x). Use this to sketch a possible graph of f (x).

The information contained in the graph of f'(x) is enough to determine the sign and
concavity of f (x). It is possible to clearly pick out intervals where f'(x) is positive or
negative. Also, f''(x) is positive wherever f'(x) in increasing, and f''(x) is negative
wherever f'(x) is decreasing. Putting this information together generates the following
chart:
This can be graphed in the following way: