Your Algebra Homework Can Now Be Easier Than Ever!

Math 171 Review for the Final Exam Solutions

1. Find the limits (4 points each)

Solutions (a) The limit represents an indeterminate
form . Therefore we can keep only the leading terms in the
numerator
and in the denominator of the fraction .
. Because x tends to positive infinity we have
whence .

(b) The limit represents an indeterminate form . As
always when we deal with such an indeterminate form we will first
find the limit of the natural logarithm of our expression . We have
, and the limit is an indeterminate
form because the first factor tends to ∞ and the second one -
to ln 1 = 0. To apply the L'Hospital's rule we have to write the last
limit as an indeterminate form or . In our case it is easier to write
it in the form , namely, . According to the
L'Hospital's rule we will now differentiate the numerator and the
denominator and look at the following limit (when differentiating the
numerator we combine the chain rule and the quotient rule).

Therefore .

(c) We have here an indeterminate form . First we bring it to
the form , .

Next we use the L'Hospital's rule to obtain the following limit

It is still an indeterminate form so we apply the L'Hospital's rule
once again and get the limit

(d)

This is an indeterminate form and we apply the L'Hospital's rule.

2. Find the derivatives of y = f(x) with respect to x (4 points each)

Solutions.

(a) f(x) = cos (sin (x + 1)). We apply the chain rule to get

f'(x) = -sin (sin (x + 1)) cos (x + 1):

(b) . By the quotient rule combined with the chain rule
we have

We simplify the last expression by multiplying both numerator and
denominator by .

(c) . By the product rule combined with the chain
rule we have

If we recall the formula

then we can also write our answer as

(d) 5y2 + sin y = x2. We apply implicit differentiation to get

Solving it for we get

3. Find the integrals (5 points each)

Solutions

(a)

If we notice that x is proportional to the derivative of 3x2 + 1 we can
use the following substitution, u = 3x2 +1. Then , du = 6xdx,
and therefore . The integral becomes

Applying the power rule we get

(b)

Notice that sec2 x equals to the derivative of tan x and therefore it is
convenient to use the substitution u = tan x. Then du = sec2 xdx and
our integral becomes

(c) We apply the power rule and the Newton - Leibnitz formula to get

(d)

We use the substitution u = t3 + 1. Then du = 3t2dt whence
6t2dt = 2du. If t = -1 then u = 0, and if t = 0 then u = 1, we will
change the limits of integ ration accordingly .

4. Use a linear approximation to estimate the following values (5 points
each)

Solutions In both cases we use the formula for linear approximation

L(x) = f(a) + f'(a)(x - a)

where point a should satisfy two (informal) conditions

• a should be close to x
• the values of f (a) and f0(a) should be easy to compute.

(a) In this case we take whence . We also take
a = 64 and x = 63. Then f(a) = 4, , and x - a = -1. The
corresponding linear approximation to is

(b) Remember that we do not use the degree measure in calculus. In
our case we take

Then

Therefore the value of the linear approximation is

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of July 16th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2018, Algebra-Answer.Com.  All rights reserved.