Calculus I
Study Guide
MA3310 Calculus I
Study Guide
EXERCISE 1.1 (1.5 HOURS)
Assessment Preparation Checklist:
To prepare for this assessment, you should:

Read Chapters 1 and 2, pp. 32–56, of your textbook, Calculus. Chapter 1 focuses on functions
used in calculus, and Chapter 2 focuses on limits.

Review the lesson for this module that explains various types of functions and how to work with
limits.
Title: Review of Functions and Limits
1. Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformation to
sketch the graph of the function p(x) = 3 sin (2x – π/3). Use a graphing utility only to check your
work.
2. Let f  x  
x2  5x  6
x2  2 x
a. Calculate lim f ( x), lim f ( x), lim f ( x), and lim f ( x) .
x 0
x 0
x 2
x 2
b. Does the graph of f have any vertical asymptotes? Explain.
Submission Requirements:
Submit your answers in a Microsoft Word document. Name the document
MA3310_StudentName_Module1_Exercise.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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MA3310 Calculus I
Study Guide
LAB 1.1 (1.5 HOURS)
Assessment Preparation Checklist:
To prepare for this lab, you should:

Read Chapters 1 and 2, pp. 32–56, of your textbook, Calculus. Chapter 1 focuses on functions
used in calculus, and Chapter 2 focuses on limits and continuity.

Review the lesson for this module that explains various types of functions and how to work with
limits.
Title: Functions and Limits
Solve the following problems, providing detailed steps wherever required.
1. A stone is thrown vertically upward from the ground at a speed of 40 m/s at time t = 0. Its
distance d (in m) above the ground (neglecting air resistance) is approximated by the function
f(t) = 40t–5t².
Determine an appropriate domain of the function. Identify the independent and dependent
variables.
2. Let g(x) =
x
and h(x) = x³ − 4x² + 6. Calculate the value g(h(x)), for x=0,1,2,3,4,5.
6x
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MA3310 Calculus I
Study Guide
4. Graph the following function.
g x  3 x  5  2

5. Find an approximate value of sec   . Round your answer to the nearest hundredth.
4
 
6. Solve the equation 2 sinx 2  0 . Give a general formula for all the solutions by using angle(s)
in the interval [0,2π] and adding multiples of some integer k.
7. Find the exact value of each of the remaining trigonometry functions of θ.
cos   
12
, θ in quadrant II
13
8. A rock is dropped off the edge of a cliff and its distance s (in feet) from the top of the cliff after t
seconds is s(t) = 16t2. Assume the distance from the top of the cliff to the water below is 1936ft.
a. When will the rock strike the water?
b. Make a table of average velocities and approximate the velocity at which the rock strikes
the water.
Time Interval
10,11 10.5,11 10.9,11 10.99,11 10.999,11
Average Velocity
 ft / sec 
9. Use the graph of h in the given figure to identify the following values, if they exist.
a. h(5)
b.
lim h  x 
x 5
c. h(7)
d.
e.
lim h  x 
x 7
lim h  x 
x 8
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MA3310 Calculus I
10. Let g(t)=
t  100
t  10
Study Guide
.
a. Complete the following table by computing values of g(t) for the given values of t.
t
g(t)
99.900
99.990
99.999
100.100
100.010
100.001
b. Make a conjecture about the value of lim
x 100
t  100
t  10
.
11. Find the given limit.
lim (-x2+3x-6)
x 1
12. Find the given limit.
lim
h 0
7
7h  1  1
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MA3310 Calculus I
Study Guide
13. Use the following function to answer questions (a) and (b) below.
4  x, x  4
f x  
 x  1, x  4
a. Find lim f (x) and lim f (x) .
x  4
x  4
b. Does lim f (x) exist? If so, what is it? If not, why not?
x 4
14. Evaluate the following limit.
7
lim 21  x 
3
x 6
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MA3310 Calculus I
Study Guide
Submission Requirements:
Submit your answers in a Word document. Name the document
MA3310_StudentName_Module1_Lab.docx, replacing StudentName with your name.
Evaluation Criteria:
Your submission will be evaluated against the following criteria:

Did you include appropriate steps, graphs, or rationales to determine the answers to questions
wherever required?

Did you correctly answer each question?
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All Rights Reserved.
[7]
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MA3310 Calculus I
Study Guide
READ AND BEGIN PROJECT (0.5 HOURS)
Refer to “Project: Problem-Solving Skills” for a detailed description of the project.
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MA3310 Calculus I
Study Guide
APPENDIX A: HANDOUTS AND WORKSHEETS
PROJECT TOPIC LIST
Topic: Numerical differentiation
Topics and skills: Derivatives, calculator
While the rules of differentiation allow us to compute the derivative of just about any function, there
are practical situations in which these rules cannot be used. For example, in some applications, a
relationship between two variables may be given as a set of data points, but not as a formula. In
situations like this, the rate of change of one variable with respect to the other (that is, the derivative)
might be needed, but the rules do not apply to sets of data. This project focuses on methods for
approximating the derivative of a function at a particular point.
Backward and Forward Difference Formulas
Assuming the limit exists, the definition of the derivative f (a)  lim
h 0
f (a) 
f (a  h)  f (a)
h
f (a  h)  f (a)
implies that
h
(1)
for h near 0. If h > 0, then (1) is referred to as a forward difference formula and if h < 0, (1) is a
backward difference formula. The geometry of these formulas is shown in Figure 1.
1. Why do you think (1) is called the forward difference formula if h > 0 and a backward difference
formula if h < 0?
2. Let f (x) =
x
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MA3310 Calculus I
Study Guide
a. Find the exact value of f '(4).
b. By equation (1), f (4) 
f (4  h)  f (4)

h
4h 2
. Therefore we estimate f '(4) by
h
4h 2
for values of h near 0. Complete columns 2 and 5 of table given below1
h
calculating
and describe
h
4h 2
behaves as h approaches 0.
h
4h 2
h
Error
h
0.1
–0.1
0.01
–0.01
0.001
–0.001
0.0001
–0.0001
4h 2
h
Error
Table 1
3. The accuracy of an approximation is given by
Error = |exact value – approximate value|.
Use the exact value of f’(4) in part (a) to complete columns 3 and 6 in Table 1. Describe the
behavior of the errors as h approaches 0.
Centered Difference Formulas
Another formula that is used to approximate the derivative of a function at a point is the centered
f (a)  lim
difference formula (CDF)
4. Again consider f (x) =
h 0
f (a  h)  f (a  h)
2h
(2)
x.
a. Graph f near the point (4, 2) and let h = ½ in the centered difference formula. Show the line
whose slope is computed by the centered difference formula and explain why the formula
approximates f '(4).
b. Use the centered difference formula to approximate f '(4) by completing Table 2.
h
Approximation
Error
0.1
0.01
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MA3310 Calculus I
Study Guide
h
Approximation
Error
0.001
0.0001
Table 2
5. Use the CDF (2) and a table similar to Table 2 to find a good approximation to f'(0) for
f(x) = (1 + x)–1.
6. Use the CDF (2) and a table similar to Table 2 to find a good approximation to f'(π/6) for
f(x) = sin x.
7. Table 3 gives the distance f(t) fallen by a smokejumper t seconds after she opens her chute.
a. Use the forward difference formula (1) with h = 0.5 to estimate the velocity of the skydiver
at t = 2 s.
b. Repeat part (a) using the centered difference formula (2).
t (seconds)
f(t) (feet)
0
0
0.5
4
1.0
15
1.5
33
2.0
55
2.5
81
3.0
109
3.5
138
4.0
169
Table 2
Computer Rounding Error
Using difference approximations to approximate derivatives with a computer or calculator is prone to
rounding errors. These errors occur when a calculator rounds a number before using it in an arithmetic
calculation. Such rounding may lead to remarkably inaccurate results.
8. Consider the function f (x) = x10.
a. Use analytical methods to find the exact value of f'(1).
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MA3310 Calculus I
Study Guide
b. Use the forward difference formula to approximate f'(1) using values of h = 10–2, 10–3, and
10–4. What do you observe?
c. Compute approximations to f'(1) using h = 10–n for n = 5, 6, 7, …, 15 What do you observe?
In Step 8c, you should find that for small enough values of h, the approximations to f'(1)
eventually are 0, which is clearly a bad estimate. Here is why this error occurs. Suppose h =
10–14. The calculator rounds f(1 + 10–14) to 1 and therefore the forward difference formula
becomes
f (1  1014 )  f (1)
14
10
, which is estimated to equal 1  1 or 0.
1014
d. The remedy to rounding errors in this situation is to use small—but not too small—values of
h. Based on the approximations computed in parts (b) and (c), what is a good approximation
to f'(1)?
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MA3310 Calculus I
Study Guide
Topic: Elasticity in economics
Topics and skills: Derivatives
Economists apply the term elasticity to supply, demand, income, capital, labor, and many other variables
in systems with input and output. In a few words, elasticity describes how changes in the input to a
system are related to changes in the output. And because elasticity involves change, it also involves
derivatives. In this project we investigate the idea of elasticity as it applies to demand functions. It’s a
common experience that as the price of an item increases, the number of sales of that item generally
decreases. This relationship is expressed in a demand function.
1. Suppose a gas station has the linear demand function D(p) = 1200 – 200p (Figure 1). According
to this function, how many gallons of gas can the gas station owners expect to sell per month if
the price is set at $4 per gallon?
2. Evaluate D’(p) and show that the demand function is decreasing. Explain why demand functions
are usually decreasing functions.
3. Suppose the price of a gallon of gasoline (Steps 1 and 2) increases from $3.50 to $4.00 per
gallon; call this change Δp. What is the resulting change in the number of gallons sold, call it ΔD?
(Note that the change is a decrease, so it should be negative.)
4. Now express the answer to Step 3 in terms of percentages: What is the percent change in price,
Δp/p, when it increases from $3.50 to $4.00 per gallon? What is the resulting percent change in
the number of gallons sold, ΔD/D? (Note that the percent change is negative.)
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MA3310 Calculus I
Study Guide
5. The elasticity in the demand is the ratio of the percent change in demand to the percent change
in price; that is, E 
D / D
. Compute the elasticity for the changes in Steps 3 and 4 (it should
p / p
be negative).
6. The elasticity is simplified by considering small changes in p and D. In this case we use the
definition of the derivative and write
E  lim
p 0
D / D
D  p  dD p
 lim
   dp D .
p / p
p  0 p  D 
Now the elasticity is a function of p. Show that for the gasoline demand function the elasticity is
E(p)  
p
.
6p
7. The elasticity may be interpreted as the percent change in the demand that results for every
one percent change in the price. For example if E(p) = –2, a one-percent increase in price
produces a two-percent decrease in demand. If the price of gasoline is p = $4.50 and there is a
3.5% increase in the price, what is the elasticity and the corresponding percent change in the
number of gallons sold?
8. Graph the gasoline demand elasticity function for 0 ≤ p < 6.
9. When –∞ < E < –1, the demand is said to be elastic. When –1 < E < 0, the demand is said to be
inelastic. When E = –∞, the demand is perfectly elastic and when E = 0 the demand is perfectly
inelastic. Essential goods such as basic foods tend to have inelastic demands; discretionary
items, such as electronic equipment have elastic demands. Explain the meaning of these terms
in this context.
10. For what prices is the gasoline demand function elastic and inelastic?
11. The demand for processed pork in Canada is described by the function D(p) = 286 – 20p1. Graph
the demand function, compute the elasticity, and graph the elasticity. For what prices is the
demand function elastic and inelastic?
12. Show that the general linear demand function D(p) = a – bp, where a and b are positive real
numbers, has a decreasing elasticity for 0 ≤ p < a/b. Show that for the general linear demand
function, E = –1, when p = a/2b.
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MA3310 Calculus I
Study Guide
13. Not all demand functions are linear. Compute the elasticity for the exponential demand function
D(p) = ae–bp, where a and b are positive real numbers.
14. Show that the demand function D(p) = a/pb, where a and b are positive real numbers, has a
constant elasticity for all positive prices.
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