Download Homework 6 - Select Problems

Math 400
Calculus I
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Homework 6 - Select Problems
Homework 6 - Assigned Problems
Section 3.4
3.4.11
Use Theorem 3.11 to evaluate the following limits.
lim
x→0
tan(7x)
sin x
Solution.
tan(7x)
sin(7x) 1
= lim
x→0 sin x
x→0 cos(7x) sin x
sin(7x)
1
1
= lim
x→0
1 cos(7x) sin x
sin(7x) 7x
1
1
= lim
·
·
x→0
1
7x cos(7x) sin x
1
7x
sin(7x)
·
= lim
x→0
7x
cos(7x) sin x
sin(7x)
1
x
= lim
· lim
· 7 lim
x→0
x→0 cos(7x)
x→0 sin x
7x
=1·1·7
=7
lim
3.4.19
Find
dy
dx
for the following functions.
y = sin x cos x
Solution.
We have to use the product rule.
y 0 = cos x · cos x + sin x(− sin x)
= cos2 (x) − sin2 (x)
= cos (2x)
By an identity.
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ARC
Math 400
Calculus I
Homework 6 - Select Problems
ARC
Section 3.6
3.6.41
Calculate the derivative of the following function.
√ tan e 3x
Solution.
We have four “layers” of applying the chain rule for this problem.
√ √
1
0
2
y = sec e 3x · e 3x · (3x)−1/2 · 3
2
For the above, the function between each dot is a separate derivative of an “inner”
function.
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