Articles

1.6E: Exercises - Mathematics


Section Exercise

Verbal

Exercise 1.6.1

How do you solve an absolute value equation?

Answer:

Isolate the absolute value term so that the equation is of the form (|A|=B). Form one equation by setting the expression inside the absolute value symbol, (A), equal to the expression on the other side of the equation, (B). Form a second equation by setting (A) equal to the opposite of the expression on the other side of the equation, (−B). Solve each equation for the variable.

Exercise 1.6.2

How can you tell whether an absolute value function has two x-intercepts without graphing the function?

Exercise 1.6.3

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

Answer:

The graph of the absolute value function does not cross the x-axis, so the graph is either completely above or completely below the x-axis.

Exercise 1.6.4

How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?

Exercise 1.6.5

How do you solve an absolute value inequality algebraically?

Answer:

First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.

Algebraic

Exercise 1.6.6

Describe all numbers (x) that are at a distance of 4 from the number 8. Express this using absolute value notation.

Exercise 1.6.7

Describe all numbers (x) that are at a distance of (dfrac{1}{2}) from the number −4. Express this using absolute value notation.

Answer:

(|x+4|= frac{1}{2})

Exercise 1.6.8

Describe the situation in which the distance that point (x) is from 10 is at least 15 units. Express this using absolute value notation.

Exercise 1.6.9

Find all function values (f(x)) such that the distance from (f(x)) to the value 8 is less than 0.03 units. Express this using absolute value notation.

Answer:

(|f(x)−8|<0.03)

For the following exercises, solve the equations below and express the answer using set notation.

Exercise 1.6.10

(|x+3|=9)

Exercise 1.6.11

(|6−x|=5)

Answer:

({1,11})

Exercise 1.6.12

(|5x−2|=11)

Exercise 1.6.13

(|4x−2|=11)

Answer:

({frac{9}{4}, frac{13}{4}})

Exercise 1.6.14

(2|4−x|=7)

Exercise 1.6.15

(3|5−x|=5)

Answer:

({frac{10}{3},frac{20}{3}})

Exercise 1.6.16

(3|x+1|−4=5)

Exercise 1.6.17

(5|x−4|−7=2)

Answer:

({frac{11}{5}, frac{29}{5}})

Exercise 1.6.18

(0=−|x−3|+2)

Exercise 1.6.19

(2|x−3|+1=2)

Answer:

({frac{5}{2}, frac{7}{2}})

Exercise 1.6.20

(|3x−2|=7)

Exercise 1.6.21

(|3x−2|=−7)

Answer:

No solution

Exercise 1.6.22

(|frac{1}{2}x−5|=11)

Exercise 1.6.23

(| frac{1}{3}x+5|=14)

Answer:

({−57,27})

Exercise 1.6.24

(−|frac{1}{3}x+5|+14=0)

For the following exercises, find the x- and y-intercepts of the graphs of each function.

Exercise 1.6.25

(f(x)=2|x+1|−10)

Answer:

((0,−8)); ((−6,0)), ((4,0))

Exercise 1.6.26

(f(x)=4|x−3|+4)

Exercise 1.6.27

(f(x)=−3|x−2|−1)

Answer:

((0,−7)); no x-intercepts

Exercise 1.6.28

(f(x)=−2|x+1|+6)

For the following exercises, solve each inequality and write the solution in interval notation.

Exercise 1.6.29

(| x−2 |>10)

Answer:

((−infty,−8)cup(12,infty))

Exercise 1.6.30

(2|v−7|−4geq42)

Exercise 1.6.31

(|3x−4|geq8)

Answer:

(−dfrac{4}{3}{leq}xleq4)

Exercise 1.6.32

(|x−4|geq8)

Exercise 1.6.33

(|3x−5|geq-13)

Answer:

((−infty,− frac{8}{3}]cupleft[6,infty ight))

Exercise 1.6.34

(|3x−5|geq−13)

Exercise 1.6.35

(|frac{3}{4}x−5|geq7)

Answer:

((-infty,-frac{8}{3}]cupleft[16,infty ight))

Exercise 1.6.36

(|frac{3}{4}x−5|+1leq16)

Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

Exercise 1.6.37

(y=|x−1|)

Answer:

Exercise 1.6.38

(y=|x+1|)

Exercise 1.6.39

(y=|x|+1)

Answer:

For the following exercises, graph the given functions by hand.

Exercise 1.6.40

(y=|x|−2)

Exercise 1.6.41

(y=−|x|)

Answer:

Exercise 1.6.42

(y=−|x|−2)

Exercise 1.6.43

(y=−|x−3|−2)

Answer:

Exercise 1.6.44

(f(x)=−|x−1|−2)

Exercise 1.6.45

(f(x)=−|x+3|+4)

Answer:

Exercise 1.6.46

(f(x)=2|x+3|+1)

Exercise 1.6.47

(f(x)=3|x−2|+3)

Answer:

Exercise 1.6.48

(f(x)=|2x−4|−3)

Exercise 1.6.49

(f(x)=|3x+9|+2)

Answer:

Exercise 1.6.50

(f(x)=−|x−1|−3)

Exercise 1.6.51

(f(x)=−|x+4|−3)

Answer:

Exercise 1.6.52

(f(x)=frac{1}{2}|x+4|−3)

Technology

Exercise 1.6.53

Use a graphing utility to graph (f(x)=10|x−2|) on the viewing window ([0,4]). Identify the corresponding range. Show the graph.

Answer:

range: ([0,20])

Exercise 1.6.54

Use a graphing utility to graph (f(x)=−100|x|+100) on the viewing window ([−5,5]). Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

Exercise 1.6.55

(f(x)=−0.1|0.1(0.2−x)|+0.3)

Answer:

x-intercepts:

Exercise 1.6.56

(f(x)=4 imes10^{9}|x−(5 imes 10^9)|+2 imes10^9)

Extensions

For the following exercises, solve the inequality.

Exercise 1.6.57

(|−2x− frac{2}{3}(x+1)|+3>−1)

Answer:

((−infty,infty))

Exercise 1.6.58

If possible, find all values of (a) such that there are no x-intercepts for (f(x)=2|x+1|+a).

Exercise 1.6.59

If possible, find all values of (a) such that there are no y-intercepts for (f(x)=2|x+1|+a).

Answer:

There is no solution for a that will keep the function from having a y-intercept. The absolute value function always crosses the y-intercept when (x=0).

Real-World Applications

Exercise 1.6.60

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and (x) represents the distance from city B to city A, express this using absolute value notation.

Exercise 1.6.61

The true proportion (p) of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Answer:

(|p−0.08|leq0.015)

Exercise 1.6.62

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable (x) for the score.

Exercise 1.6.63

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using (x) as the diameter of the bearing, write this statement using absolute value notation.

Answer:

(|x−5.0|leq0.01)

Exercise 1.6.64

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is (x) inches, express the tolerance using absolute value notation.


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The Three-Moment Equation

The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.

Consider three points on the beam loaded as shown.

From proportions between similar triangles:
$dfrac> = dfrac - h_3>$

$t_ <3/2>= dfrac<1>left[ A_2ar_2 + (frac<1><2>M_2L_2)(frac<2><3>L_2) + (frac<1><2>M_3L_2)(frac<1><3>L_2) ight]$

Multiply both sides by 6
$dfrac<1>left(dfrac<6A_1ar_1> + M_1L_1 + 2M_2L_1 ight) + dfrac<1>left( dfrac<6A_2ar_2> + 2M_2L_2 + M_3L_2 ight)$
$= 6left( dfrac + dfrac ight)$

Combine similar terms and rearrange

If E is constant this equation becomes,

If E and I are constant then,

For the application of three-moment equation to continuous beam, points 1, 2, and 3 are usually unsettling supports, thus h1 and h3 are zero. With E and I constants, the equation will reduce to

Factors for the three-moment equation
The table below list the value of $6Aar/L$ and $6Aar/L$ for different types of loading.


Fiches d'Exercices de Maths le Plus Populaires cette Semaine

Les enseignants en éducation spécialisée, les éducateurs pour adultes, les tuteurs et les services de tutorat, les collèges et universités, les professeurs de mathématiques du secondaire et les étudiants eux-mêmes utilisent également Mathslibres régulièrement. Les enseignants en éducation spécialisée apprécient particulièrement le fait que nous décomposons les compétences en mathématiques et fournissons des options en gros caractères. Les apprenants adultes apprécient le format simple et épuré proposé par nos feuilles de travail.

Les tuteurs et les entreprises de tutorat utilisent nos feuilles de calcul pour réduire leurs coûts et se concentrer sur l'apprentissage des élèves. Les enseignants du secondaire, des collèges et des universités ont souvent besoin de ressources de rattrapage pour les étudiants qui poursuivent des études supérieures, et nombre d’entre eux choisissent Mathslibres.

Les élèves qui mettent en pratique leurs compétences en mathématiques avec nos feuilles de calcul pendant les vacances scolaires conservent leurs compétences en mathématiques pour les prochains trimestres. Parce que nous fournissons des corrigés, les étudiants sont capables de s'auto-évaluer et d'utiliser les commentaires immédiats fournis par un corrigé pour analyser et corriger les erreurs dans leur travail.

Nous travaillons dur pour nous assurer que notre site Web fonctionne bien et nous avons les meilleures feuilles de calcul mathématiques. En règle générale, si vous utilisez une connexion haut débit, vous verrez notre contenu en moins d'une seconde.


Money Math

Money is used on a daily basis, so it's essential for children to know how to use it responsibly. That is why the Learning Library supplies teachers, parents and tutors with a generous supply of money math resources for preschoolers and up.

The hundreds of printable worksheets range from lessons on basic money terms to word challenges. Some assignments take students ""shopping"" at coffee shops or a sports store to learn practical money word problems. Young students learn how to count money and how cents translates to dollars. A variety of coloring worksheets familiarize small students with how money looks.

For interactive lessons, the resource library possesses multiple online games where kids can sort money or learn decimal division with pennies. There is an assortment of hands-on activities and games that utilize real money for practice. Kids can compete to count to a dollar first with a racing game or build their own construction paper piggy bank.

There are many step-by-step guides that are created by professional educators. Popular lessons such as Money Math, Show Me the Money and Add It Up! Counting Money make teaching streamlined, yet entertaining. An abundance of other guided lessons, interactive books and more are easy to access from the Learning Library to teach students to be money masters.


1.6E: Exercises - Mathematics

Remember the rule in the following way. Always start with the ``bottom'' function and end with the ``bottom'' function squared. Note that the numerator of the quotient rule is identical to the ordinary product rule except that subtraction replaces addition. In the list of problems which follows, most problems are average and a few are somewhat challenging.

Click HERE to see a detailed solution to problem 1.

Click HERE to see a detailed solution to problem 2.

Click HERE to see a detailed solution to problem 3.

Click HERE to see a detailed solution to problem 4.

Click HERE to see a detailed solution to problem 5.

Click HERE to see a detailed solution to problem 6.

Click HERE to see a detailed solution to problem 7.

Some of the following problems require use of the chain rule.

Click HERE to see a detailed solution to problem 8.

Click HERE to see a detailed solution to problem 9.

Click HERE to see a detailed solution to problem 10.

Click HERE to see a detailed solution to problem 11.

Click HERE to see a detailed solution to problem 12.

Click HERE to see a detailed solution to problem 13.

Click HERE to see a detailed solution to problem 14.

Click HERE to see a detailed solution to problem 15.

Click HERE to see a detailed solution to problem 16.

Click HERE to see a detailed solution to problem 17.

Click HERE to see a detailed solution to problem 18.

Click HERE to see a detailed solution to problem 19.

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :


A large laboratory has four types of devices used to determine the pH of soil samples. The laboratory wants to determine whether there are differences in the average readings given by these devices. The lab uses 24 soil samples having known pH in the study, and randomly assigns six of the samples to each device. The soil samples are tested and the response recorded is the difference between the pH reading of the device and the known pH of the soil.

Based on your intuition, is there evidence to indicate any difference among the mean differences in pH readings for the four devices?

Run an analysis of variance to confirm or reject your conclusion of part (1). Use (alpha = 0.05) .

Compute the p-value of the F test in part (2).

What conditions must be satisfied for your analysis in parts (2) and (3) to be valid?


The Product Rule

and in this quite simple case, it is easily seen that the derivative of a product is NOT the product of the derivatives. Although this naive guess wasn't right, we can still figure out what the derivative of a product must be. Remember: When intuition fails, apply the definition. Consider

Now we apply the trick of adding zero, in the form of u ( x + h ) v ( x ) - u ( x + h ) v ( x ) to the numerator, and after performing some minor algebra,

because u ( x ) is differentiable at x and therefore ontinuous.

A good way to remember the product rule for differentiation is ``the first times the derivative of the second plus the second times the derivative of the first.'' It may seem non-intuitive now, but just see, and in a few days you'll be repeating it to yourself, too.

Another way to remember the above derivation is to think of the product u ( x ) v ( x ) as the area of a rectangle with width u ( x ) and height v ( x ). The change in area is d ( uv ), and is indicated is the figure below.

As x changes, the area changes from the area of the red rectangle, u ( x ) v ( x ), to the area of the largest rectangle, the sum of the read, green, blue and yellow rectangles. The change in area is the sum of the areas of the green, blue and yellow rectangles,

In the limit of dx small, the area of the yellow rectangle is neglected. Algebraically,

``Neglecting'' the yellow rectangle is equivalent to invoking the continuity of u ( x ) above. This argument cannot constitute a rigourous proof, as it uses the differentials algebraically rather, this is a geometric indication of why the product rule has the form it does.


Lessons and Worksheets

CLOTHING STORE WORD PROBLEMS

With this random worksheet, have your students answer simple questions about spending money.

THE BAKERY WORD PROBLEMS

With this random worksheet, have your students answer simple questions about spending money.

THE VALUE OF MONEY

The Case of the Broken Piggy Bank. Students list the values of a quarter, dime, nickel, and penny in dollar form. Then they find the total monetary value of a set of money, and calculate change received on a purchase. Includes lesson plan, student lesson, and printable worksheet.

BACK TO SCHOOL

Practice money skills with a back to school theme.

SAVING FOR FRECKLES

A story about spending and saving money and using it wisely. Includes reading comprehension worksheet.

SPENDING MONEY FOOD SHOPPING LESSONS

GET A DISCOUNT

Teach your students the concept of a discount, while reinforcing basic math skills.` Learn to be a good consumer.

SHOPPING LIST

Practice using a food shopping list with this printable worksheet lesson.

MATH FOR GROCERY SHOPPING LESSON PLAN

Teaching lesson on grocery shopping math, with a focus on produce. Students determine grocery costs. Includes lesson and worksheets.

NEEDS OR WANTS

For budgeting or otherwise, every good consumer need to understand the difference between needs and wants.

FRACTIONS AND MONEY

How many pieces in a chocolate candy bar? Students identify the fractions including the numerator and denominator, and learn to multiply fractions. Includes lesson plan, student lesson, and printable worksheet.

How many quarters in one dollar? Students learn about reciprocals, converting fractions to solve, and dividing fractions.

Students learn to convert fractions, decimals, and percents.

GROCERY STORE RECEIPT

Practice calculating receipt totals and taxes while reinforcing basic consumer math skills such as adding and percentages.

READING A RESTAURANT MENU

Practice reading items on a simple restaurant menu and answering the spending money questions.

RESTAURANT CHECK

Practice calculating restaurant check charge totals and taxes.

TIPPING LESSON

Practice calculating a tip at a restaurant. Tipping the waiter or waitress.

ESTIMATING YOUR GROCERIES

Students practice their math skills at estimation while shopping for groceries.

GROCERY SHOPPING WITH COUPONS

Students use the coupons and answer the questions about grocery shopping with coupons. An introductory lesson on using coupons and understanding discounts. Basic consumer money math.

COMPARISON SHOPPING WORD PROBLEMS

Learn about consumer comparison shopping and comparing prices with this lesson and word problem worksheet. Compare different products.

NUTRITION LESSON

When spending money at the grocery, one should keep in mind meeting their daily nutritional needs. With this lesson, students' practice their understanding of nutrition.

COOKING WITH RECIPES

A lesson on cooking with recipes and learning how to convert servings.

MEASUREMENT: LIQUID CAPACITY -- BUYING APPLE JUICE

Students learn about measurement and converting units of liquid capacity including: fluid ounces, cups, pints, quarts, and gallons.

CHANGES TO THE FOOD PYRAMID

Students learn about changes to the food pyramid, health, and nutrition with this reading comprehension lesson.

SALES INVOICE OFFICE SUPPLIES WORKSHEET

Worksheet to practice using a sales invoice.

BLANK MONEY ORDER FORM

A sample money order form for your money lesson or practice filling out a money order.

WHICH COSTS MORE?

PAYING BILLS

Students practice paying bills, writing out checks, and updating their bank check register. Learn real-life money skills including making consumer payments for monthly bills and purchases.

SALES TAX - PAYING TAXES

Lessons and worksheets on paying taxes, including sales tax. Calculate sales tax, using percents to find total cost.

BUYING A CAR LOAN INTRODUCTION

Read the car advertisements and answer the questions about taking a car loan with this loan worksheet. Down payments, finance charges.

MAKING SPENDING DECISIONS ABOUT MONEY

In this lesson, students will learn ways to become mindfully deliberate about their spending choices using the DECIDE process. They will use this model to help them with making a decision for a major spending purchase.

RENTING AN APARTMENT

Read the apartment advertisements and answer the questions about choosing an apartment to rent. Focus is on rent comparison.

CHARITIES, DONATING, AND SHARING MONEY

Lessons on charities, donating, sharing, and giving money.

CREDIT CARDS

Credit cards, credit, and paying interest. Credit card consumer math skills.

BUDGETING MONEY

Worksheets and Lessons with a budgeting money theme. Learn about budgeting money issues to help with spending lessons.

ADDITIONAL LESSONS

Gambling — Risky Business
Students learn about the costs and benefits of gambling.

Spending Plans
Students learn about making spending plans and the importance of saving.

Allowances and Spending Plans
Students learn an introduction to allowances.

Comparison Shopping – Needs and Wants
This lesson introduces students to the basics of comparison shopping.

Additional Spending Money Lessons
Additional spending money related lessons.

TEACHING SPENDING MONEY AND CONSUMER MATH IDEAS

Students prepare a meal and learn about saving money.

A teaching lesson plan idea on money values and shopping.

ADDITIONAL SPENDING MONEY, SHOPPING, AND CONSUMER MATH RESOURCES

Problem Solving Using Shopping Lists - Students use their problem-solving skills and try to stay within their budget as they buy and create their own shopping list.

Grocery Shopping for a Family Profile - Students create menus and shopping lists based on dietary restrictions of a family.

CONSUMER SPENDING AND SAVING MONEY INFORMATION AND ADVICE

Spending Tips and Saving Money Information
Learn about the basics of saving money, and spending money wisely. Includes saving money information, tips, and advice.

Information on personal finance, careers, and health.

SUGGESTIONS OR NEED HELP?

Do you have a recommendation for an enhancement to this spending money lesson page, or do you have an idea for a new lesson? Then leave us a suggestion.

To teach and learn money skills, personal finance, money management, business, careers, and life skills please go to the Money Instructor home page.


Math Games

Math games are a fun and engaging way for students to practice and build fluency with specific math skills. However, well-designed games also provide opportunities for students to foster reasoning, problem solving, and strategic thinking.

Before perusing the links below, I highly recommend watching this 3-minute video from Dan Finkel where he elaborates on the following three criteria that make a math game fun and worth playing:

  1. Student choice
  2. Mathematics is the engine of the game
  3. Easy to learn and quick to play – perfect for classroom use

Investigations Game Center

The Game Center contains online versions of many of the games from the Investigations 3 curriculum. The game center can be accessed in English and Spanish.

Math For Love Game Collection

Math For Love has curated a collection of the best math games they know. Games can be filtered by grade level and topic.

Richmond School District Math Games

This game collection is curated by Janice Novakowski. For each game there is a downloadable resource document along with a video demonstrating how to play the game.

Here’s an example video from the site of the classic game Circles and Stars:

Games for Young Minds

Games for Young Minds is where math educator Kent Haines shares his recommendations for math games to play with children. You can sign up for his weekly newsletter or peruse the archives. There are sections for board games and free games.