Problem Sets for the Harvard Precalculus

Table of Contents

• Lesson 1: Points & Coordinates
• Lesson 1.5: Functions
• Lesson 2: Slopes
• Lesson 3: Linear Functions
• Lesson 3.5: Linear Function Moving
• Lesson 4: Linear Intersections
• Lesson 5: Exponential Functions
• Lesson 6: Function Operations
• Lesson 7: Inverse Functions
• Lesson 8: Logarithms
• Lesson 10: Quadratics
• Lesson 11: Quadratic Intersections
• Lesson 12: Finding Quadratics
• Lesson 13: Polynomials
• Lesson 14: Trigonometry Definitions
• Lesson 15: Trigonometry Functions
• Lesson 16: Trigonometric Identities

Lesson 1: Points & Coordinates

1. 
   What are the coordinates of points a, and b?
2. Draw Points at 7,2, and 3,5.
3. What is the distance between:

7,2 and 3,2

3,5 and 3,2

7,2 and 3,5

• Give coordinates for points c,d,e,f,g,h.
• Draw points at: -5,8 8,-4 -7,-6
• What is the distance between a and h
• What is the slope of the line between: a,b c,a e,f g,h d,e

Real Estate Price Discussion

The price of real estate is based on the total acreage contained within the property lines. Map makers use plat maps to describe property lines. Each property line has a distance between corners and a bearing angle. The bearing angle starts with a vertical direction (N or S) then the angle (for example 35°) and then a horizontal angle (E or W). The bearing angle N35°W would be as the diagram shows. Bearing angles are always measured from the NS line and always are from 0° to 90°.

A triangular piece of property begins at the origin and goes 5' at N36.87°E then it goes 10' at N53.13°W and then it returns to the origin. Where are the corner points of this property? What is its perimeter?

• Consider the triangular property described by points a, d, and e. Can you describe it using bearing angles? How about a, c, d, h, e? What is its perimeter?

Lesson 1.5: Functions

The United States federal highway Administration collects data on domestic motor fuel consumption. The table below lists the average fuel efficiency for cars in the U.S. measured in miles per gallon.

1. Graph the data table.
2. Explain why the efficiency is a function of the year (why is year the independent variable).
3. If F(year) is the fuel efficiency function:
1. F(1985) = ?
2. F(1992) = ?
3. F(?) = 18.20
4. F(?) = 21.69
4. The empirical function describes just the data that is collected but does not consider the values that were not presented. What is the domain and range of the empirical fuel efficiency function?
5. The actual function is the values that could have been measured even if no measure was taken. What do you think is the domain and range of the actual function? Why?
6. Text Section 1.1 pg. 8 probs. 5,7,8
7. Text Section 1.2 pg. 14 probs. 1,3
8. Text Section 1.3 pg. 22 probs. 1,6,7
9. Text Section 1.4 pg. 27 probs. 1,3

Lesson 2: Slopes

You are going on a job interview in Canada. You will need some money when you get there and that money should be Canadian. The teller hands you this chart to help you decide how much money to exchange:

1. Graph these pairs.
2. What is the slope of these pairs? Do they line up?
3. You have $75 that you want to exchange. How much Canadian money will you get?
4. Your hotel room costs 100 Canadian dollars a night, how much is that in American dollars?
5. Express this function using an equation. What is the domain and range of the function?
6. Use the equation to compute how many American dollars you need to acquire a $20,000 car in Canada.
7. Use the equation to compute how many Canadian dollars a Canadian needs to acquire a $20,000 car in the U.S.
8. Text Section 1.6 pg. 46 probs. 1 , 13, 15.
9. Text Section 2.1 pg. 72 probs. 1,2,6,12
10. Text pg. 54 probs. 4, 12

Lesson 3: Linear Functions

Part 1: Algebraically

1. Find the intercepts of the line through these points with the specified slopes:

Point                                Slope

(3,-5)                               -2

(2,6)                                -3

(6,7)                                2

(6,8)                                1/3                                     

• Find the slope and intercepts of the line through these pairs of points:

(-1,1)                               (3,9)

 (3,-5)                               (-1,7)

 (2,6)                                (5,3)

 (6,7)                                (2,7)

 (6,8)                                (-2,4)                               

Section 2.3 pg. 94 probs. 1, 5, 7, 10

Part 2: Pratical Problems.

• Section 2.1 pg. 73 probs. 2, 11, 16.
• Section 2.2 pg. 83 probs. 10,11,12,18
• Freezing is 32 degrees Farenheight but 0 degrees Celsius. Boiling is 212 degrees Farenheight but 100 degrees Celsius. What is the Formula to translate Celsius to Farenheight? What is the Formula to translate Farenheight to Celsius? Why do we care?
• Section 2.3 page 80 probs. 15, 16, 19.

Lesson 3.5: Linear Function Moving

1. Let f(x) = 2x, What is its intercept?
   What is the intercept of f(x)+1? f(x)+3? f(x)-2?
2. Graph g(x) = -x-2. Graph g(x)+3. How did adding 3 change the graph? What would g(x)-2 do to the graph?
3. Section 4.1 pg. 205 probs. 18,19,20
4. Assume that a store sells 5000 packets of cheese at $2 and 3000 packets at $3. Assuming that demand is linear with price graph the number of packets sold vs the price. Put number of packets on the x axis and price on the y axis.
5. Assume that the cheese becomes more popular and people are willing to pay $1 more for the same packet of cheese. Graph the new line between price and quantity sold.
   At the new level of demand how many packets will sell at $5?
6. Assume there is a scandal and now people want it only for $2 less than the original price (in question 3). How many packets will be sold for $1?

Lesson 4: Linear Intersections

Intersect Algebraically and graphically these lines:

y=2x-3                               y=-x+3

y=2x+3                               y=3x+6

slope 2 through (2,6)                through (5,4) and (6,2)

y=-2x+5                              slope 4 through (2,7)

y = ½x+3                             slope 1 through (-2,3)               

1. Section 2.3 Pg 95 Problems 14,18,20
2. You can sell watercolor paintings for $20 a piece. You want your profits to be 20% of your costs. Making watercolor paintings costs $40 fixed costs and $15 for each painting. How many paintings do you need to sell before you are making your profit?
3. At the concert, you can sell 6 copies of your poster at $100 and 96 copies at $10 a piece. If you have time to make 42 copies before the concert then how much should you charge to sell all of them? If you charged $1 more would you make more money?
4. Your fame grows and your posters are more in demand. Now you can sell 19 copies at $100 and 109 copies at $10 a piece. Now how much do you charge to sell 43?
5. If making 60 posters would earn you more than $100 extra above what you are making from the 43 posters, you would make 70 posters. Will you make 70 posters? How about 91?

Lesson 5: Exponential Functions

1. Section 3.1 pg. 120 prob. 1,2,4
2. Calculate: 3⁴, (1/3)⁴, (2/3)⁴, 3^-4, (1/3)^-4, (2/3)^-4
3. Express as simply as possible: (a²)(a³), (a³)(a^-2), a^-2b^-2, a³b^-3, (d²)³,d²³, (n^-3k⁶)^2/3
4. Sec 3.2 Pg. 133 Probs. 1,2,3,5
5. Probs. 8,7
6. Probs. 14-19
7. Probs 24

Lesson 6: Function Operations

1. Section 4.1 pg. 201 prob. 1,3,10,11.13,14,16
2. Section 4.3 pg 226 prob. 1,3
3. Section 4.5 pg 249 prob. 2,3,4
4. Section 6.1 pg 352 prob 1,2,5
5. prob. 32,

Lesson 7: Inverse Functions

1. Section 6.2 pg 363 prob. 1,4,7,8,9,12,13
2. Section 6.3 pg. 372 prob. 1,6
3. Demand functions tell you how much of a product such as cheese you can sell at a specified price. They have negative slopes, why? Interpret the inverse function of demand. Discuss who might want to know it. What will its slope be? Why?

Lesson 8: Logarithms

1. Section 3.4, pg 153 prob 1,2,4,7,13
2. prob. 15,19,20,21
3. Section 3.6 pg 167 prob 1,2,3,4,5
4. prob 11,13,21
5. Section 3.7 pg 176 prob 5,12,19,20

Lesson 10: Quadratics

1. Section 4.4 Pg 239 probs 1,2,3
2. probs 5-8
3. probs 12, 13
4. probs 28-30
5. probs 42,43
6. If you charge $20 your store sells 500 crystals a year. When you raise the price by $1 you sell 10 less, if you lower the price by $1 you sell 10 more. To produce the crystals costs $1000 fixed costs and $5 of materials for each crystal. How much do you charge to maximize revenue (total money the store collects)? How do you charge to maximize your profits (revenue-manufacturing costs)? Whats the maximum price that earns you more than $5000?
7. You can sell 30 paintings at $20 and reducing the price by $2 allows you to sell 10 more paintings You want your profits to be 20% of your costs. Making watercolor paintings costs $40 fixed costs and $15 for each painting. How much do you charge for your 20% profit? (note this problem is hard)

Lesson 11: Quadratic Intersections

1. Find the intersection points between these curves - Graphically and Algebraically:

Lesson 12: Finding Quadratics

Click here to find the formula for a quadratic through 3 points

1. find the quadratic through these triples:

1,2	2,3	3,6
-1,2	2,-1	3,-6
1,5	-2,-3	3,6
1,-2	-2,4	3,-6

2. pg 243 prob 38

Lesson 13: Polynomials

Lesson 14: Trigonometry Definitions

1. How many radians is 60^o, 30^o, 15^o, 135^o, 400^o
2. Show where each of these lie on the unit circle.
3. Section 5.2 pg 320 prob 6.
4. How many degrees in radians, /4 radians, 3/8 radians, 1 radian, 7 radians.
5. Show approximately where each of these lie on the unit circle.
6. What is the sin, cos and tan of the angle that the line y= x makes with the line y=0.
7. What is the sin, cos and tan of the angle that the line y= 2x makes with the line y=0.
8. What is the sin, cos and tan of the angle that the line y= ½x makes with the line y=0.
9. What is the sin, cos and tan of the angle that the line y= -x makes with the line y=0.
10. What is the sin, cos and tan of the angle that the line y= -2x makes with the line y=0.
11. What is the sin, cos and tan of the angle that the line y= -2x makes with the line x=0.
12. What is the sin, cos and tan of the angle that the line y= x makes with the line x=0.
13. What is the sin, cos and tan of the angle that the line y= ½x makes with the line x=0.

Lesson 15: Trigonometry Functions

1. Build these tables:

-/2

-/4

0

/4

/2

3/4

5/4

3/2

7/4

2

Sin(x)

Cos(x)

2sin(x)

Sin(2x)

-cos(x)

1. Use the tables to graph sin(x), cos(x), 2sin(x), sin(2x), - cos(x).
2. Give the amplitude, frequency, and altitude for each of the functions below
   (each x box is /4 radians):
   (a) (b) (c)
3. Graph each of these functions:

• sin(2)
• -cos(-)
• 3cos(+) +2
• 2sin(2+/2)-2

1. Section 5.5 pg. 299 prob. 5, 10,11,18,20,25

Lesson 16: Trigonometric Identities

1. Get all of these in terms of the sin and cos of using trigonometric identities:

• sin(2)
• -cos(-)
• 3cos(+) +2
• 2sin(2+/2)-2
• tan(3)

1. Use trigonometric identities to calculate the sin, cos and tan for each of these:

• 7/12
• 17/12
• -5/12
• /12
• -19/12
• 23/12

© David B. Sher 1996

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Funded by the NSF grant: Development of Course Materials to Promote Collaborative Learning through Interactive Animation for Mathematics, National Science Foundation Course and Curriculum Development Program.

On Equipment Provided by the NSF Grant 9451689: Interactive Animations for Mathematics and Computer Science, National Science Foundation Instrumentation and Laboratory Improvement Program