Math 180 Calculus 1 Worksheets, Study notes of Calculus

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Math 180 Calculus 1 Worksheets

Department of Mathematics, Statistics, and Computer Science

University of Illinois at Chicago

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Contents

• Weekly worksheets
  • 1 Functions and slopes
  • 2 Properties of limits...................................................................................................
  • 3 Computing limits
  • 4 Computing more limits
  • 5 Limits at infinity and asymptotes
  • 6 Continuity
  • 7 Definition of the derivative.......................................................................................
  • 8 Computing derivatives..............................................................................................
  • 9 Product and quotient rules, derivatives of trigonometric functions..........................
  • 10 The chain rule, and derivatives as rate of change
  • 11 Derivatives of logarithmic, exponential, and inverse trigonometric functions
  • 12 Related rates
  • 13 Maxima and minima, what the derivative tells us
  • 14 Properties of graphs of functions..............................................................................
  • 15 Graphs of functions and introduction to optimization
  • 16 Optimization
  • 17 The mean value theorem and introduction to l’Hˆopital’s rule
  • 18 L’Hˆopital’s rule and antiderivatives
  • 19 Riemann sums
  • 20 Sigma notation and more integration.......................................................................
  • 21 The fundamental theorem of calculus
  • 22 Properties of the definite integral.............................................................................
  • 23 Vectors in the plane
  • 24 Dot product in the plane..........................................................................................
• Review worksheets
  • 1 Review of worksheets 1-10........................................................................................
  • 2 Review of worksheets 11-16......................................................................................
  • 3 Review of worksheets 1-24........................................................................................
• Index

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1 Functions and slopes

Keywords: functions, domain, graphing, slope, secant lines

1. In your own words, describe what is the domain and range of a function.
2. Consider the following conditions on a function f :

· the domain of f is all real numbers

· f (x) > 0 for any real number x

· f (4) = f (4)

Answer the questions below using di↵erent functions in each part.

(a) Give an example of a function that satisfies the first condition.

(b) Give an example of a function that satisfies the second condition.

(c) Give an example of a function that satisfies the third condition.

!! (d) Give an example of a function that satisfies all three conditions.

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3. Consider the function f (x) = sin(x).

(a) Draw the graph of f from 0 to 4⇡ below. Make sure to label the axes and unit

lengths.

(b) Find the slopes of the secant lines between the given pairs of points below.

i. (0, f (0)) and (⇡/ 2 , f (⇡/2))

ii. (0, f (0)) and (⇡, f (⇡))

iii. (0, f (0)) and (3⇡/ 2 , f (3⇡/2))

iv. (0, f (0)) and (2⇡, f (2⇡))

v. (0, f (0)) and (3⇡, f (3⇡))

vi. (0, f (0)) and (4⇡, f (4⇡))

(c) Generalize from the above to find the slope of the secant line between (0, f (0))

and (k⇡, f (k⇡)) for any positive or negative integer k = 0, 1 , 2 , 3 ,....

!! (d) Besides the pairs of points in part (c) above, how many other pairs of points exist

with the same property? Write down three other such pairs.

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3. On Monday afternoon Olive gets on the Kennedy Expressway at North Avenue and

gets o↵ at Taylor Street, driving a distance of 3 miles on the highway. There is a 60

MPH sign at the Randolph Street exit, which is 2 miles down from the North Avenue

ramp.

(a) On Tuesday she gets a speeding ticket, because video cameras on the highway

indicated that she was on the highway for 2 minutes and 30 seconds. If the

recording is to be trusted, what was her average speed over the 3 mile section?

(b) On Wednesday Olive goes to court and claims it took her 1 minute and 20 seconds

to drive the first two miles, where the speed limit was 90 MPH, and 1 minute and

10 seconds to drive the third mile. If Olive is to be trusted, what was her average

speed

i. over the first two miles? ii. over the last mile?

(c) On Thursday the county judge decides that Olive deserves the ticket if her speed

is more than 60 MPH when she passes the sign. Is Olive’s argument enough to

prove that her speed was not more than 60 MPH when she passed the sign? Why

or why not?

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3 Computing limits

Keywords: limits, graphing, one-sided limits

1. Let f, g be two functions with lim

x!c

f (x) = 2 and lim

x!c

g(x) = 6. Showing all your steps,

simplify the following limits.

(a) lim

x!c

[8g(x)]

(b) lim

x!c

[5f (x) + 9g(x)]

(c) lim

x!c

g(x)

(d) lim

z!c

[f (z)/ 3 z]

!! (e) lim

x!c+

[2f (x 2)]

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4. Compute the following limits or state that they do not exist.

(a) lim

x! 0

|x|

x

(b) lim

x! 0

|x|

x

(c) lim

x! 0

|x|

x

5. With help from the question above, draw a graph of

|x|

x

on the interval [ 5 , 5]. Make

sure to label the axes and unit lengths.

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4 Computing more limits

Keywords: limits, squeeze theorem, infinite limits

1. Determine the following limits or state that they do not exist.

(a) lim

x! 1

(x 1)(x 2)

(x 1)

3

(b) lim

x! 2

x

3 5 x

2

• 6x

x

3 4 x

(c) lim

x! 1

x 1

p

x 1

2. Consider the graphs of y = x sin(1/x), y = |x|, and y = |x|, given below.

|x|

|x|

x sin(1/x)

x

y

3 2 (^1 1 2 )

1

1

Using the graph and the squeeze theorem, evaluate lim

x! 0

x sin

1

x

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5 Limits at infinity and asymptotes

Keywords: limits, limits at infinity, asmyptotes, graphing

1. (a) Evaluate the following limits.

i. lim

x!

2 x

4 x

2 8 x

ii. lim

x!+ 1

3 x

5 x

3

• 8x

5 x

5 7

iii. lim

x!

6 x

7 4 x

2

• 2

x

2 3 x + 5

iv. lim

x!+ 1

2 x

2 3 x

x

4 7

(b) Do any of the functions in the limits above contain horizontal asymptotes? If so,

explain why.

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2. Consider the function f (x) =

3 x

2 x

x

2 6 x + 5

(a) List any vertical asymptotes of f (x), and explain, using limits, why each is a

vertical asymptote.

(b) List any horizontal asymptotes of f (x), and explain, using limits, why each is a

horizontal asymptote.

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6 Continuity

Keywords: continuity, graphing, intermediate value theorem

1. On the grid below, draw a function f that is not continuous at x = 2 with lim

x! 2

f (x) = 1.

2. On the grid below, draw a function f that is defined everywhere and whose limit

lim

x! 2

f (x) is undefined.

3. Are the functions you drew above continuous at x = 2

(a) from the left?

(b) from the right?

Recall that a function is continuous from the left at x = a if lim

x!a

f (x) = f (a), and

continuous from the right if lim

x!a

f (x) = f (a).

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4. On the grid below, draw the function

f (x) =

5 if x < 0 ,

sin(⇡x) if 0 6 x 6 1 ,

x 1 if x > 1.

On what intervals is f (x) continuous? Explain your reasoning!

5. Determine whether or not f is continuous at x = a. If not, explain why.

(a) f (x) =

p

x 5 at a = 3

(b) f (x) =

x

2 4

x + 2

at a = 2