HCCS Continuity and Derivatives Questions

Math 1431Homework Assignment 5 (Written)
Limits, Continuity and Derivatives (Chapters 1 and 2)
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1. (12 points) (a) (8 points) Evaluate the limit lim
h→0
3
4+h
−
3
4
h
(b) (4 points) The limit above can be interpreted as f 0 (c), the derivative of a function evaluated at some point x = c. Identify
a point x = c and a function f (x) that matches this interpretation.
1
2. (12 points) The piece-wise function g(θ) is given by

 A sin θ + cot θ
g(θ) =

cos θ − 6θ
π
: θ ≤ π/2
: θ > π/2
(a) (6 points) Determine the value of the constant A so that g(θ) is continuous on the interval (0, π).
(b) (6 points) Using the value for A from part (a), is g(θ) differentiable on the interval (0, π)? Explain your answer.
3. (9 points) Several expressions are written below, and each one contains a function and an interval. Circle only
those expressions that satisfy the conditions of the IVT with N equal to any value in between the given function’s end point
outputs (no work need be included with this problem).
(a) f (x) = −x2 + sec x on [0, 3π]
(d) j(x) = 2x on [−2, 3]
(g) m(x) =
(b) g(x) =
1
on [−2, 2]
2 + sin x
(e) k(x) = arctan x on [−1, 1]


x2 −3x+2
x−1
: x 6= 1

−1
: x=1
on [0, 4]
(c) h(x) = −x2 + sec x on [0, π/4].
(f) `(x) =
(h) p(x) = any continuous function on (−16, 8]
2
|x − 3|
on [3, 2022]
x−3
(i) q(x) = x2/3 on [−3, 3]
4. (12 points) Use the diagram below to sketch a graph of a function y = f (x) with the following properties (make
sure your graph is clearly drawn so that these features are present):
a) The domain of f (x) is [−4, −2) ∪ (−2, 4]
b) The range of f (x) is (−∞, ∞)
c) f (x) has a removable discontinuity at x = 0.
d) lim− f (x) = −1 and 3 = f (2) = lim+ f (x)
x→2
x→2
0
e) f (3) > 0
f) f (x) is continuous at x = −3 but f 0 (−3) DNE
3
2
1
-4
-2
2
4
1
-2
3
5. (12 points) Several expressions are written below, and each expression represents a real number. Arrange these
expressions in order from greatest to least; for example, you might think (a) > (b) > (c) > (d) > (e) > (f), but this example
is wrong so don’t use it as your answer. (No work needs to be included with this problem.)
24 + 10
t
t→∞ 6 + 22
t
(a) lim
(b)

d p
5 − x2
dx
>
24 + 10
t
t→0 6 + 22
t
(c) lim
x=1
>
>
3

00
(d) −5×2
>
sin(5θ)
θ→0 sin(20θ)
(e) lim
>
(f)

d 
x cos x
dx
x=π
6. (10 points) One of the world’s least-regarded companies, Chegg Inc.1 , runs a website that some critics and reviewers
lambaste as a “waste of money” run by “con artists.” This company claims that students who pay for their services are more
likely to succeed in college courses, but such claims are dubious at best. Shown below is a graph y = f (x) that displays
a typical Chegg user’s performance on graded work in a Math course over a four month period. Their running average is
displayed along the vertical y-axis with the horizontal x-axis displaying time.
Running Average out of 100
100
90
Q
80
P
y=f(x)
70
60
50
R
40
1 Month
2 Months
3 Months
4 Months
(a) (3 points each) Naming the labeled points so that P = (p, f (p)), Q = (q, f (q)) and R = (r, f (r)), arrange the numbers
f 0 (p), f 0 (q) and f 0 (r) in order from least to greatest:
< < Work for this part of the problem can be shown by sketching tangent lines at the points P, Q and R labeled above. These tangent lines need only be hand-drawn, based on the provided graph (as more detailed sketches would have required a formula for f (x) that is not provided). (b) (1 point) Based on the given graph, what is the students’ actual course grade (approximately)? That is, approximate the value of lim f (x). x→4− 1 see, for example, the following: Link 1, Link 2, Link 3, Link 4, Link 5, Link 6 4 7. (12 points) Some information about two functions, f (x) and g(x), are provided in the table below; for this question it is assumed that the compositions f ◦ g and g ◦ f are well-defined functions. x 3 4 5 f (x) g(x) 1 5 2 0 4 6 f 0 (x) g 0 (x) 4 3 2 2 14 1 Based on the information provided, answer the following questions.  0 (a) f ◦ g (3) =  0 (b) g ◦ f (5) =   (c) Find the slope of the line tangent to h(x) = f x2 + 1 when x = 2. 8. (12 points) Several statements are presented below; mark each one as either T (“TRUE”) or F (“FALSE”). No work need be included for this problem. a) If f 0 (2) exists then lim f (x) = f (2) = lim f (x) = lim f (x). x→2 x→2+ x→2− b) The tangent line for y = f (x) at the point (c, f (c)) has as its equation y = f (c) + f 0 (c)(x − c). √ c) The two functions r(t) = t2 − t+sin2 t+4 and `(t) = t2 −t1/2 −cos2 t+π have the same derivative.  d) lim θ→0 1 θ csc θ 2 = 1. 5 9. (12 points) The set of points that satisfy the equation xy + x = 2y 2 trace out the curve shown in the image below. As indicated, this curve passes through the points (1, 1) and (0, 0). (a) Use implicit differentiation to find the slope of the line tangent to this curve at the point (1, 1) or explain why the slope is undefined. (b) Use implicit differentation to find the slope of the line tangent to this curve at the point (0, 0) or explain why the slope is undefined. 6 10. (7 points) Let’s do a derivative “in reverse!” That is, instead of handing you a function f (x) and asking you to compute f 0 (x), let’s start with the derivative and ask you to figure out which function was differentiated. To this end, suppose we have f 0 (x) = π + 2x − sin x + 2x−3 . Which, if any, of the following functions could equal the original f (x)? (a) f (x) could equal 2 − cos x − 6x−4 (b) f (x) could equal πx + x2 + cos x − x−2 + 42 − √ (c) f (x) could equal πx + x2 + cos x + x−2 + 2 (d) f (x) could equal πx + x2 + sin x − 6x−4 + √ √ 2022 17 + 2 + 1 π (e) None of the above 7