Written Homework 10
Math 1200
Instructions: Submit your work on Gradescope bySunday, October 27, 2024 at 11:59
pm. You must show all work.
1. Find thex-coordinates of the local extrema ofh(x) =e
x
(x
3
−4x+ 4).
2.Consider the functionf(x) =
(
√
x+B x >4
Ax
2
e
x
x≤4
,
whereAandBare constants. Find
values ofAandBso thatf(x) is differentiable at all points in its domain.
1
3.Letf(x) =
Ax
B
x
2
+ 5
,whereAandBare constants. Find the values ofAandBso that
the tangent line off(x) atx= 1 has the equationy= 3x+ 1.
4. Compute the following limits:
(a) lim
x→0
x
2
e
x
x
2
−9x
(b) lim
x→5
x
2
+ 25
4x+ 8
(c) lim
x→3
+
2
x
−8
x
2
−6x+ 9
(d) lim
x→∞
5
x
−8x
2
+ 3x−5
9x
8
−10(5
x
) + 2
2
5. Find the absolute extrema ofh(x) =x
√
2−x
2
on [−
√
2,
√
2].
6. Find thex-coordinates of the inflection points off(x) =
4
x
(ln(4))
2
−32x
2
+ 3x+ 5
3
7.The table below gives values of the functionf(x) and its derivativef
′
(x). The graph of
a piece-wise linear functiony=g(x) is also provided.
x23568
f(x)4-320-1
f
′
(x)-354-17
12345678910
1
2
3
4
5
6
7
8
9
10
y=g(x)
x
y
Compute the following. If any of the values does not exist, writeDNE.
(a)p
′
(2),wherep(x) = 4f(x)−5g(x)
(b)q
′
(3),whereq(x) =x
2
f(x)
(c)r
′
(5),wherer(x) =f(g(x))
(d)j
′
(5),wherej(x) =g(f(x))
(e)s
′
(6), wheres(x) =
5g(x)−4x
8−f(x)
(f) lim
x→2
(f(x))
2
−16
g(x)−6
4
