MATH 221: Calculus I
HW 5 Solutions
Summer 2013 (M. Stapf)
Due: July 9th
Instructions: (10 points) Write up all solutions on another paper. Please clearly label each question, and
circle or box your final answers.
Exponential and Logarithmic Functions
1. Simplify 3e2 ln(5) to a constant.
= 3(5)2 = 75.
2. Simplify ln(e2 · (1 + e0 )) − ln(2) to a constant.
= 2 ln(e) + ln(2) − ln(2) = 2.
3. Simplify 4 ln
p
e(x+1) to a polynomial.
=4
4. Simplify
x+1
2
ln(e) = 2x + 2.
1
ln(x10 ) + ln(x−3 ) − ln(x2 ) to a polynomial. Note: This polynomial may look strange.
2
= 0.
5. Simplify e(x
2
+1)
· (ex )2 to a single exponential.
= ex
6. Simplify
e3x
1/2
(e4x )
2
+2x+1
.
to a single exponential.
= ex .
√
7. Simplify 6 log( x) − log(x) to a single logarithm.
= log(x2 )
8. Express 17x in the form of ekx for some constant k.
= eln(17)x .
9. Express e · 2x in the form of ekx+b for some constants b and k.
= e · eln(2)x = eln(2)x+1 .
10. Express 33x+3 in the form of Cekx for some constants C and k.
= 33 · 33x = 27 · e3 ln(3)x .
1
Differentiating Exponentials and Logarithms
11. Differentiate e3x .
d 3x
e = 3e3x .
dx
12. Differentiate ln(17x) and ln
1
x . Do you notice any interesting relationships between these two?
2
d
1
1
ln(17x) =
17 = .
dx 17x
x
d
1
1 1
1
ln
x 1 · = .
dx
2
2
x
2x
13. Differentiate ln(x2 + 1).
2x
d
ln(x2 + 1) = 2
.
dx
x +1
14. Differentiate e1−x .
d 1−x
d
e
= e1−x ·
(1 − x) = −e1−x .
dx
dx
15. Differentiate
√
ex − 1.
d √ x
1
d x
ex
e −1= √ x
·
(e − 1) = √ x
.
dx
2 e − 1 dx
2 e −1
x
16. Differentiate e(e ) .
x
x
d x
d (ex )
e
= e(e ) ·
e = e(e +x).
dx
dx
√
17. Simplify, then differentiate 4x ln( x).
√
d
4x ln( x) = 2x ln(x)
dx
= 2 ln(x) + 2x ·
= 2 ln(x) + 2.
18. Find the first and second derivatives for f (x) = x · ex
f 0 (x) = xex + ex
f 00 (x) = xex + ex + ex
= xex + 2ex .
19. Find the first and second derivatives for f (x) = ln(3x).
1
1
·3=
3x
x
1
00
f (x) = − 2 .
x
f 0 (x) =
2
1
x
1
.
ex + 1
20. Find the first and second derivatives for f (x) =
f (x) = (ex + 1)
−1
−2
f 0 (x) = − (ex + 1) · ex
i
h
−3
−2
f 00 (x) = − −2 (ex + 1) (ex )(ex ) + (ex + 1) ex
= 2 (ex + 1)
−3 2x
e
− (ex + 1)
−2 x
e .
We could also simplify further by forming a common denominator, resuling in
f 00 (x) =
e2 x − ex
.
(ex + 1)3
21. Find the first, second, and third derivatives for f (x) = x ln(x) − x.
f 0 (x) = ln(x)
1
f 00 (x) =
x
1
000
f (x) = − 2 .
x
2
22. Find the first, second, and third derivatives for f (x) = e(x ) .
2
f 0 (x) = 2xe(x
)
2
f 00 (x) = 2e(x ) + 4x2 e(x
2
2
)
2
f 000 (x) = 4xe(x ) + 8xe(x ) + 8x3 e(x
2
= e(x ) 8x3 + 12x
2
)
23. Find the line tangent to f (x) = log2 (x + 1) at x = 1.
We can find the point we desire by plugging in x = 1, thus we are looking at the point (1, 1). To find
the slope, we will need the derivative of f (x).
f 0 (x) =
Thus the desired slope is m =
1
.
ln(2)(x + 1)
1
. This results in the formula for the line,
2 ln(2)
y−1=
1
(x − 1) .
2 ln(2)
3