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2 Page calculus worksheet I need each problem worked out thoroughly so I can study each problem.
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Calculus 201 for ayphxl
Studypool Tutor jmmanley
Problem 1
Sketch the graph of f & g: f (x) = 3x , g(x) = log3 x. See the plot of the graphs below:
Figure 1: In black: f (x) = 3x . In blue: f (x) = log3 (x) = log(x)/ log(3).
Problem 2
Differentiate the following functions.
a.
f (x) = xe2x
f 0 (x) = x(2e2x ) + e2x = (2x + 1)e2x
b.
f (x) = xe−x
2
2
2
f 0 (x) = x(−2xe−x ) + e−x = (1 − 2x2 )e−x
2
c.
f (x) = (1 + x−2x )4
d
d
d
(1 + x−2x ) = 4(1 + x−2x )3 (x−2x ) = 4(1 + x−2x )3 e−2x log(x)
dx
dx
dx
d
d
f 0 (x) = −8(1 + x−2x )3 e−2x log(x) (x log(x)) = −8(1 + x−2x )3 x−2x (x log x + log x)
dx
dx
f 0 (x) = 4(1 + x−2x )3
f 0 (x) = −8x−2x (1 + x−2x )3 (1 + log x)
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d.
f (x) =
p
6e−x + 2x
1
1 − 3e−x
f 0 (x) = (6e−x + 2x)−1/2 (−6e−x + 2) = √
2
6e−x + 2x
e.
2
f (x) =
ex
x
2
2xex2 ex
2
f (x) =
− 2 = (2 − x−2 )ex
x
x
0
Problem 3
Differentiate the following functions.
a.
f (x) = ln x8
f 0 (x) =
1 7
8
8x =
8
x
x
b.
f (x) = ln(x3 + 1)
f 0 (x) =
1
3x2
2
3x
=
x3 + 1
x3 + 1
c.
f (x) = 3x2 ln x
f 0 (x) = 6x ln x + 3x2
1
= 3x(1 + 2 ln x)
x
d.
f (x) = ln
√
x
1
1
1 1
f 0 (x) = √ (x)−1/2 = √ √ =
2x
x2
2 x x
e.
This is the same as subsection e in problem 1. Please see above.
Problem 4
Differentiate the following functions.
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a.
f (x) = ln(e4x + 3)
1
4e4x
4x
4e
=
e4x + 3
e4x + 3
f 0 (x) =
b.
2
f (x) = ln(ex + 1)
2
f 0 (x) =
1
2xex
x2
2xe
=
ex2 + 1
e x2 + 1
c.
f (x) =
f 0 (x) =
ln x
1
= ln x
x
1+e
1 + ex
1 1
1
1
ex ln x
x
−...