Written Homework 3
Math 1200
Instructions: Submit your work on Gradescope bySunday, September 8, 2023 at 11:59
pm. You must show all work.
1.Jordan and Devin run a 5 kilometer race in a straight line. LetJ(t) denote the distance,
in km, that Jordan has run aftertminutes, and letD(t) denote the distance, in km,
that Devin has run aftertminutes. The following table lists some values ofJ(t) and
D(t):
m410121620
J(t)0.81.72.13.24
D(t)1.12.52.93.74.5
(a) Compute Jordan’s average velocity on the interval [10,20]. Include units.
(b) Estimate Devin’s instantaneous velocity att= 10. Include units.
(c)
Suppose that fromt= 20 until Jordan finishes the race, Jordan runs with an
average velocity of 0.25 kilometers per minute. On the other hand, suppose Devin
runs the entire race with an average velocity of 0.2 kilometers per minute. Who
wins the race?
1
2. Shown below is a portion of the graph of the functionf(x), which has domain (0,20).
1234567891011121314151617181920
1
2
3
4
5
6
7
8
9
10
11
y=f(x)
x
y
(a) Find the following limits, or explain why they do not exist.
i. lim
x→2
f(x)
ii.f(2)
iii. lim
x→8
f(x)
iv. lim
x→5
f(x)
v. lim
x→5
−
f(x)
vi. lim
x→5
+
f(x)
vii. lim
x→10
−
f(x)
viii. lim
x→10
+
f(x)
ix.lim
x→−10
+
f(−x)
x. lim
x→2
x
2
f(x)
xi. lim
x→5
+
p
f(x) + 2
xii. lim
x→8
−
f(x)
x
2
+ 4
(b) Find all the pointsx=cin the domain off(x) where lim
x→c
−
f(x)̸=f(c).
(c)Find the following limits.Hint: All three limits exist, and each has a different
value.
i. lim
x→2
f(f(x))ii. lim
x→8
f(f(x))iii. lim
x→13
f(f(x))
2
3. Sketch the graph of a functionf(x) satisfying the following properties:
•The domain off(x) is the interval [−5,5].
•lim
x→−3
+
f(x) =f(−3)
•lim
x→−3
f(x) does not exist.
•lim
x→−1
f(x) = 2
•f(−1) =−1
•The slope of the secant line on the interval [−1,1] is
3
2
.
•lim
x→3
+
f(x) = 2−lim
x→3
−
f(x)
•f(x) is decreasing on the interval [3,5].
−5−4−3−2−11234567
−5
−4
−3
−2
−1
1
2
3
4
5
x
y
3
4. Below is a portion of the graph of a functionf(x).
1234567
−1
1
2
3
4
y=f(x)
x
y
In addition,g(x) is defined to be
g(x) =
5x−3x <2
3x+ 3 2≤x≤5
4x−2x >5
.
Find the following limits or determine that they do not exist.
(a) lim
x→5
f(x)
(b) lim
x→5
g(x)
(c) lim
x→2
(3g(x)−x
2
f(x))
(d) lim
x→5
(f(x))
3
(g(x))
1/2
(e) lim
x→1
p
g(x) + 2
4−2f(x)
4
5. Consider the piecewise functionk(x) below.
k(x) =
(
A(x
2
−5)
1/3
ifx≤5
2x+Bifx >5
Find values of the constantsAandBso that
•lim
x→2
k(x) = 4 and
•lim
x→5
k(x) exists.
5
6.Compute the following limits, or briefly explain why they do not exist. Simplify your
answers.
(a) lim
x→4
x
2
+ 2x−24
x
2
−16
(b) lim
h→0
1
4+h
−
1
4
h
(c) lim
x→−5
(x+ 5)
8
−2x−10
x+ 5
(d) lim
h→0
4−
√
16 +h
h
2
−4h
6
