MATH 251 Portland Community College Calculus 1 Worksheet

Description

__________________________________________________________________________________________
1.
Determine the value of each of the following limits (if they exist).
Use the method of your choice, be sure to show all work. If you are using a theorem from your textbook,
then you must state it by name.
(8pt)
𝑙𝑖𝑚
a.
𝑥→∞
𝑙𝑖𝑚
b.
𝑥 2 −3𝑥+2
3𝑥−8
𝑥 2 −3𝑥+2
𝑥→2 2𝑥 2 −8
__________________________________________________________________________________________
2.
Neatly and carefully construct a coordinate grid (such as provided below). On your coordinate grid, sketch
a graph of a function 𝑦 = 𝑓(𝑥) that satisfies all the following conditions:
(10pt)
a. lim 𝑓(𝑥) = 2
𝑥→5
b.
c.
lim 𝑓(𝑥) = −4
𝑥→∞
lim 𝑓(𝑥) = 6
𝑥→ −5−
d.
e.
lim 𝑓(𝑥) = ∞
𝑥→ −5+
lim 𝑓(𝑥) = ∞
𝑥→−∞
3.
Use the following graph of a function, 𝑓(𝑥) to answer each of the following multiple-choice questions.
(12pt)
I.
The derivative, 𝑓′(𝑥), is equal to 0 when
a. x = -3, 1
II.
d. x = 0
b. (0.7, 4)
c. (-2.8, 4)
d. (-1, 2)
The derivative, 𝑓′(𝑥), is increasing on the interval(s)
a. (-3, 3)
IV.
c. x = -1, 2
The derivative, 𝑓′(𝑥), is positive on the interval(s)
a. (-5, -1)
III.
b. x = -5, 0.7, 4
b. (-5, 1)
c. (-1, 2)
d. (2, 5)
c. x = -1, 4
d. x = -2
The 2nd derivative, 𝑓′′(𝑥), is equal to 0 when
a. x = 1
b. x = -5, 0.7, 4
___________________________________________________________________________________________
3
4.
Let 𝑦 = √𝑥 5 . se the power rule to find 𝑦′. Show detailed steps.
(5pt)
__________________________________________________________________________________________
5.
Let 𝑦 = (2 ln 𝑥)3 .
(5pt)
a.
The given 𝑦 is a composition of functions 𝑓(𝑔(𝑥)) where
𝑓(𝑥) = ________________ and 𝑔(𝑥) = _________________ .
b.
Show detailed steps how you would find 𝑦′
__________________________________________________________________________________________
6.
The value of functions 𝑓(𝑥) and 𝑔(𝑥), as well the value of their derivatives 𝑓′(𝑥) and 𝑔′(𝑥)
(16pt)
are provided for some specific x values in the table below:
𝑥
𝑓(𝑥)
𝑓′(𝑥)
𝑔(𝑥)
𝑔′(𝑥)
2
1
2
0
-1
4
-2
4
2
-2
6
-1
1
3
-2
8
4
-2
4
-1
Use the above table to determine each of the following:
a.
𝑎(𝑥) = 𝑓(𝑥) − 𝑔(𝑥). Find 𝑎′(4).
b.
𝑏(𝑥) = 𝑓(𝑥)𝑔(𝑥). Find 𝑏′(6).
Note: the bonus problem at the end of the exam is related to this question.
c.
𝑐(𝑥) = 𝑓(𝑔(𝑥)). Find 𝑐′(8).
d.
𝑑(𝑥) = [𝑓(𝑥)]−1 . Find 𝑑′(2).
7.
Application of Implicit Differentiation
(16pt)
A hot air balloon is rising vertically at a rate
of 100 meters per minute.
A camera located 50 meters away from the
balloon launch site is filming the balloon.
Note that as the balloon rises, the angle
between the camera and the ground must be
increasing.
Your task:
Determine the rate at which the angle, 𝜃, is increasing when the balloon is 150 meters above the ground.
a.
How is the angle, 𝜃, is related to the height of the balloon? (Based on the picture; think trig).
b.
Use implicit differentiation to differentiate your answer to part a. with respect to time.
𝑑𝜃
Solve for .
𝑑𝑡
Note: your result should have a
𝑑ℎ
𝑑𝑡
term and furthermore, you have been given that
𝑑ℎ
𝑑𝑡
= 100.
c.
Determine the angle of the camera when the balloon is 150 meters off the ground.
d.
Use your results to b. and c. to determine the rate at which the angle, 𝜃, is changing when that
balloon is 150 meters off the ground.
8.
Given 𝑓(𝑡)
= 2−𝑡 ,
(15pt)
a.
Calculate 𝑓′(𝑡) then evaluate 𝑓′(1)
b.
Calculate 𝑓′′(𝑡) then evaluate 𝑓′′(1)
c.
At 𝑡 = 1 is 𝑓 increasing, decreasing, or neither?
What information from your answers to a. or b. lead you to this conclusion?
d.
At 𝑡 = 1 is 𝑓 concave up, concave down, or neither?
What information from your answers to a. or b. lead you to this conclusion?
e.
At 𝑡 = 1 is 𝑓′ increasing, decreasing, or neither?
What information from your answers to a. or b. lead you to this conclusion?
___________________________________________________________________________________________
9.
Consider the curve given by the equation 𝑥 4 + 𝑦 3 = 3𝑥 2 𝑦.
(10pt)
a.
𝑑𝑦
Use implicit differentiation and find an equation for 𝑑𝑥 .
(Your final answer may contain both x and y)
b.
Give the equation of the tangent line to this curve at the point (2,2).
10.
(3pt)
One of the following is the graph of a function, 𝑓(𝑥), another is the graph of it’s derivative, 𝑓′(𝑥), and
another is the graph of it’s 2nd derivative 𝑓′′(𝑥).
Identify which graph is 𝑓(𝑥), which graph is 𝑓′(𝑥), and which graph is 𝑓′′(𝑥).
8
6
4
2
0
-2 0
-4
-6
4
6
2
4
2
0
2
4
6
-2
8
0
2
4
6
8
0
-4
,
Graph 1
-2
-6
,
0
2
4
-4
Graph 2
Bonus: (worth +3)
Consider problem 6b. Provide the equation of the tangent line to 𝑏(𝑥) when 𝑥 = 6.
Graph 3
6
8

Purchase answer to see full
attachment