Workshop #12
Professor D. Olles
1. Sketch the graoh of a function whose derivative is zero.
2. Sketch the graoh of a function whose derivative is a non-zero constant.
3. Sketch the graoh of a function whose derivative is a constant multiple of
x.
4. Differentiate the following functions:
a. f (x) = 2ex − x2e
b. f (x) =
√
7
x2 +
6
−9e
x2
4−3x
c. f (x) = 3x
2 +x
5. Find any points at which f (x) =
4x−8
ex
has a horizontal tangent line.
√
6. Find the equation of the tangent line to the curve f (x) = x x that is
parallel to 3x − y = −1
7. Find the equation of the tangent line to each curve at the given point.
a. f (x) =
ex
1+ex
4 x
at x = 0.
b. f (x) = x e at x = ln 2.
x
8. Find a formula for the nth derivative of f (x) = ex .
1
Solutions
1. Sketch the graoh of a function whose derivative is zero.
2. Sketch the graoh of a function whose derivative is a non-zero constant.
3. Sketch the graoh of a function whose derivative is a constant multiple of
x.
4. Differentiate the following functions:
a. f (x) = 2ex − x2e
f 0 (x) = 2ex − 2ex2e−1
b. f (x) =
√
7
6
−9e
x2
x2 +
f (x) = x2/7 + 6x−2 − 9e
2 −5/7
x
− 12x−3 − 0
7
2
12
= 5/7 − 3
x
7x
f 0 (x) =
4−3x
c. f (x) = 3x
2 +x
f 0 (x) =
=
(3x2 + x)(−3) − (4 − 3x)(6x + 1)
(3x2 + x)2
−9x2 − 3x − (24x + 4 − 18x2 − 3x)
(3x2 + x)2
=
−9x2 − 3x − 21x + 18x2 − 4
(3x2 + x)2
=
9x2 − 24x − 4
(3x2 + x)2
5. Find any points at which f (x) =
f 0 (x) =
=
4x−8
ex
has a horizontal tangent line.
ex (4) − (4x − 8)ex
(ex )2
4ex − 4xex + 8ex
(ex )2
=
ex (12 − 4x)
(ex )2
=
12 − 4x
ex
2
mt = 0 =⇒ f 0 (x) = 0 for some x
0 = 12 − 4x
x=3
4
4(3) − 8
= 3
f (3) =
e3
e
4
3, 3
e
√
6. Find the equation of the tangent line to the curve f (x) = x x that is
parallel to 3x − y = −1
3x − y = −1 =⇒ y = 3x + 1 =⇒ mt = 3
f (x) = x · x1/2 = x3/2
3 1/2
x
2
3√
3=
x
2
√
2= x
f 0 (x) =
x=4
√
f (4) = 4 4 = 4(2) = 8
y − y1 = mt (x − x1 )
y − 8 = 3(x − 4)
y − 8 = 3x − 12
y = 3x − 4
7. Find the equation of the tangent line to each curve at the given point.
a. f (x) =
ex
1+ex
at x = 0.
f (0) =
f 0 (x) =
e0
1
1
=
=
1 + e0
1+1
2
(1 + ex )ex − ex (0 + ex )
(1 + ex )2
=
ex + e2x − e2x
(1 + ex )2
=
mt = f 0 (0) =
ex
(1 + ex )2
e0
1
1
=
=
(1 + e0 )2
(1 + 1)2
4
3
y − y1 = mt (x − x1 )
1
1
= (x − 0)
2
4
1
1
y− = x
2
4
1
1
y = x+
4
2
y−
b. f (x) = x4 ex at x = ln 2.
4
4
4
y = f (ln 2) = (ln 2) eln 2 = (ln 2) · 2 = 2 (ln 2)
f 0 (x) = 4x3 ex + x4 ex
3
4
3
4
f 0 (ln 2) = 4 (ln 2) eln 2 + (ln 2) eln 2 = 8 (ln 2) + 2 (ln 2)
y − y1 = mt (x − x1 )
4
3
4
y − 2 (ln 2) = 8 (ln 2) + 2 (ln 2) (x − ln 2)
4
3
4
4
5
y − 2 (ln 2) = 8 (ln 2) + 2 (ln 2) x − 8 (ln 2) + 2 (ln 2)
4
3
4
4
5
y − 2 (ln 2) = 8 (ln 2) + 2 (ln 2) x − 8 (ln 2) − 2 (ln 2)
3
4
4
5
y = 8 (ln 2) + 2 (ln 2) x − 6 (ln 2) − 2 (ln 2)
x
8. Find a formula for the nth derivative of f (x) = ex .
n = 1: f 0 (x) =
ex − xex
ex (1 − x)
1−x
=
=
e2x
e2x
ex
ex (−1) − (1 − x)ex
−ex − ex + xex
−ex (2 − x)
2−x
=
=
=− x
e2x
e2x
e2x
e
x
x
x
x
x
x
x
e (−1) − (2 − x)e
−e − 2e + xe
3e − xe
ex (3 − x)
3−x
n = 3: f 000 (x) = −
=
−
=
=
=
e2x
e2x
e2x
e2x
ex
x
x
x
x
x
x
e (−1) − (3 − x)e
−e − 3e + xe
−e (4 − x)
4−x
n = 4: f (4) (x) =
=
=
=− x
e2x
e2x
e2x
e
n−1
f (n) (x) = (−1)n+1 x
e
n = 2: f 00 (x) =
4