MTH 280 Calculus I – University of Phoenix Course Solution

MTH 280 Calculus I Entire Course

MTH 280 Week 1 MyMathLab Study Plan for Weekly Checkpoint

1.1 Functions and Their Graphs

o       Find the domain and range of functions.

o       Determine if graphs represent functions.

o       Graph and identify symmetries of a function and state where it is increasing and decreasing.

1.2 Combining Functions; Shifting and Scaling Graphs

o       Identify and write equations for functions that involve translations.

o       Shift the graph of a function and write an equation for the shifted graph.

o       Graph functions using transformations.

o       Identify and write equations for functions that involve scaling.

1.3 Trigonometric Functions

o       Use the arc length formula.

o   Evaluate trigonometric functions.

o       Graph and identify characteristics of trigonometric functions.

o   Use the addition formulas for sine and cosine.

o   Use the half-angle formulas for sine and cosine.

o   Solve trigonometric equations.

o   Use the law of sines and the law of cosines.

2.1 Rates of Change and Tangents to Curves

o       Find the average rate of change of a function over an interval.

o   Find the slope of the tangent line at a given point.

2.2 Limit of a Function and Limit Laws

o       Find the limit from graphs.

o   Find the limit of algebraic functions.

o   Find the limit of trigonometric functions.

o   Find the limit using the rules for limits.

o   Evaluate the limit of average rates of change.

o   Use the Sandwich Theorem.

o   Estimate limits.

MTH 280 Weekly MyMathLab Week 2 Checkpoint

2.3 The Precise Definition of a Limit

Find delta graphically.

Find delta algebraically.

Solve applications involving limits.

2.4 One-Sided Limits

Find one-sided limits graphically.

Find one-sided limits algebraically.

2.5 Continuity

Determine where functions are continuous.

Find limits involving trigonometric functions.

Find values that make a function continuous.

Use continuity and the intermediate value theorem to solve problems.

Analyze concepts involving continuous functions.

2.6 Limits Involving Infinity; Asymptotes of Graphs

Find limits using a graph of the function.

Find limits as x approaches infinity or negative infinity.

Find limits approaching a real number.

Graph rational functions.

Find and graph a function with the given conditions.

MTH 280 Week 3 MyMathLab Study Plan for Weekly Checkpoint

3.1 Tangents and the Derivative at a Point

Find the slope of the tangent line at given points.

Find the equation of the tangent line at given points.

Solve applications involving tangent lines at a point.

Find points on a curve that have desired tangent lines.

Find vertical tangents of a curve.

3.2 The Derivative as a Function

Calculate the derivatives of functions.

Find the slope of the tangent line for a function at a given value.

Find the equation of the tangent line at a given point of a function.

Use the alternative formula for derivatives to find the derivative of a function.

Find the derivative of a graphed function.

Solve applications involving derivatives.

Determine if functions are differentiable at certain points.

3.3 Differentiation Rules

Find the first and second derivatives of a function.

Find the first derivative of a function.

Find the derivatives of all orders of a function.

Find slopes and tangents of a function.

Solve applications involving differentiation.

3.4 The Derivative as a Rate of Change

Solve applications involving rate of change.

Analyze motion using graphs.

3.5 Derivatives of Trigonometric Functions

Find derivatives of trigonometric functions.

Find and graph tangent lines for given values of trigonometric functions.

Evaluate limits involving trigonometric functions.

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MTH 280 Weekly MyMathLab Week 3 Checkpoint

3.1 Tangents and the Derivative at a Point

Find the slope of the tangent line at given points.

Find the equation of the tangent line at given points.

Solve applications involving tangent lines at a point.

Find points on a curve that have desired tangent lines.

Find vertical tangents of a curve.

3.2 The Derivative as a Function

Calculate the derivatives of functions.

Find the slope of the tangent line for a function at a given value.

Find the equation of the tangent line at a given point of a function.

Use the alternative formula for derivatives to find the derivative of a function.

Find the derivative of a graphed function.

Solve applications involving derivatives.

Determine if functions are differentiable at certain points.

3.3 Differentiation Rules

Find the first and second derivatives of a function.

Find the first derivative of a function.

Find the derivatives of all orders of a function.

Find slopes and tangents of a function.

Solve applications involving differentiation.

3.4 The Derivative as a Rate of Change

Solve applications involving rate of change.

Analyze motion using graphs.

3.5 Derivatives of Trigonometric Functions

Find derivatives of trigonometric functions.

Find and graph tangent lines for given values of trigonometric functions.

Evaluate limits involving trigonometric functions.

MTH 280 Week 4 MyMathLab Study Plan for Weekly Checkpoint

3.6 The Chain Rule

Use the Chain Rule to find first derivatives.

Use the Chain Rule to find second derivatives.

Use the Chain Rule to evaluate derivatives at given values.

Solve theory problems involving the Chain Rule.

Solve applications involving the Chain Rule.

3.7 Implicit Differentiation

Use implicit differentiation to find first derivatives.

Use implicit differentiation to find second derivatives.

Work with slope, tangent lines, and normal lines.

Solve theory problems involving implicit differentiation.

Solve problems involving orthogonal curves.

3.8 Related Rates

Solve related rate equations.

Understand how rates of change relate.

Solve applications involving related rates.

3.9 Linearization and Differentials

Find the linearization of functions.

Use the approximation (1+x)^k=1+kx.

Find the change in f, the value of the estimate df, and the approximation error.

Find differential formulas that estimate changes in volume or surface area.

Solve applications involving linearization and differentials.

MTH 280 Week 1 Discussion – Graphing Functions

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

This week, we explore graphical characteristics of functions. Respond to the following:

You might have heard the term “exponential growth” used to describe the spread of the corona virus. Give an example of an exponential function. How does an exponential function differ from a logistic function? Which function do you think is an appropriate model for the spread of COVID-19? Why? Cite and give the reference for the sources you used to answer this question.

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 2 Discussion – Limits

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

This week, we learn about limits and continuity. Respond to the following:

The reading for this week includes the two classic problems of calculus: The problem of tangents, and the problem of area (quadrature). Choose one problem and research the mathematician(s) involved in its development. Cite and give references for all sources you use.

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 3 Discussion – Derivatives

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

This week, we begin to explore derivatives. Respond to the following:

Galileo’s formula (for falling objects) gives the position of a falling object at time t. Find the formula and use it to figure out how long it would take for an apple dropped from the restaurant at the top of the Space Needle to reach the ground. What would be its velocity when it hits the ground? (The restaurant is 500 feet from the ground.) Explain clearly and completely your work. Cite and give references for all sources you use.

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 4 Discussion – Chain Rule

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

This week, we learn about the chain rule and related rates. (Related rate problems are in the Week 5 reading, which is Chapter 6.) State the chain rule and give an example of its use. Explain clearly and completely your work. Cite and give references for all sources you use.

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 5 Discussion – Applications of Derivatives

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

This week, we explore various applications of derivatives. Respond to the following:

Answer any ONE of the following exercises in your text. Explain clearly and completely your work.

Exercise 6.1.17

Exercise 6.1.18

Exercise 6.3.2

Exercise 6.3.3

Exercise 6.3.6

Exercise 6.3.53

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 6 Discussion – Fundamental Theorem of Calculus

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

This week you learn about the Fundamental Theorem of Calculus. This theorem establishes the relationship between differentiation and integration and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. Respond to the following:

Who first formulated the Fundamental Theorem of Calculus, and when? Cite and give references for all sources you use.

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 7 Discussion – What is Calculus?

Post a total of 3 substantive responses over 2 separate days for full participation. This includes your initial post and 2 replies to classmates or your faculty member.

Due Thursday

Respond to the following:

Imagine that a friend is starting college and wondering whether to take calculus. The friend asks you “What does calculus have to do with the real world?” Answer the question and cite all sources that you use, including the textbook.

Due Monday

Post 2 replies to classmates or your faculty member. Be constructive and professional.

MTH 280 Week 7 – Final Exam

Question 1: A car starts from a point at 2:00 p.m. and travels north at 40 mph. Another car starts from the same point at 3:00 p.m. and travels west at 50 mph. After the second car has traveled 1 h, at what rate is the distance between the two cars changing?

60.42 mph

94.34 mph

30.17 mph

64.03 mph

Question 2: A spherical ball is measured to have a radius of 6 space c m with a possible measurement error of plus-or-minus 0.1 rm c m. Use the differentials to estimate the percentage error in computing the volume of the ball.

1%

3%

5%

10%

Question 3: A company determines a cost function of c equals 6 x squared minus 180 x plus 2000, where c is the cost (in dollars) of producing x number of items. How many items should the company manufacture to minimize the cost?

12

15

24

30

Question 4: The position of an object is given by the equation s left parenthesis t right parenthesis equals 2 x squared plus x minus 6. Find the time t at which the instantaneous velocity of the object equals the average velocity in the interval open square brackets 0 comma 3 close square brackets.

t equals 1.5 semicolon space s

t equals 3 semicolon space s

t equals 2 semicolon space s

t equals 0 semicolon space s

Question 5: Find the locations of local minimum and maximum of x to the power of 9 minus 4 x to the power of 8 using the second derivative test.

Local minimum at x equals 0, local maximum at x equals 32 over 9

Local minimum at x equals 32 over 9, no local maximum

Local minimum at x equals 32 over 9, local maximum at x equals 0

Local minimum at x equals 0, no local maximum

Question 6: In a shop, the revenue and the cost of a product are determined by R left parenthesis x right parenthesis equals 22 x and C left parenthesis x right parenthesis equals 2 x squared plus 2 x plus 1, respectively. If x represents the number of products, how many products should the shop sell to maximize the profit?

11

5

6

10

Question 7: Evaluate limit as x rightwards arrow 0 of fraction numerator x squared over denominator e to the power of x minus x minus 1 end fractionby applying L’Hôpital’s rule.

0

1

2

infinity

Question 8: Let f left parenthesis x right parenthesis equals x cubed minus x squared minus 1 and x subscript 0 equals 1. To the nearest three decimal places, find x subscript 5 using Newton’s method of approximation.

1.466

1.486

1.625

2.000

Question 9: Evaluate integral sin 2 x cos 2 x comma space d x

1 fourth cos 4 x plus C

negative 1 over 8 cos 4 x plus C

negative 1 fourth cos 4 x plus C

1 over 8 cos 4 x plus C

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Question 10: Evaluate integral subscript negative 1 end subscript superscript 1 left parenthesis t squared plus t plus 1 right parenthesis d t using the Fundamental Theorem of Calculus, Part 2.

8 over 3

negative 5 over 6

10 over 6

Question 11: Water is flowing into a tank at a rate of r left parenthesis t right parenthesis equals 3 square root of t over 2 end root cubic meters per minute. How much water entered the tank between 2 and 8 minutes?

3 cubic meters

6 cubic meters

14 cubic meters

28 cubic meters

Question 12: To the nearest two decimal places, calculate R subscript 5 for f left parenthesis x right parenthesis equals x cubed plus 1 on open square brackets 0 comma space 4 close square brackets.

44.96

65.00

33.77

96.16

Question 13: Use substitution to evaluate integral subscript 0 superscript straight pi over 2 end superscript sin 2 x square root of 4 plus 9 sin squared x end root space d x.

1 third left parenthesis 26 square root of 13 minus 16 right parenthesis

1 over 27 left parenthesis 26 square root of 13 minus 16 right parenthesis

2 over 27 square root of straight pi cubed end root

2 over 3 square root of straight pi cubed end root

Question 14: Evaluate the integral integral fraction numerator 1 plus tan x over denominator 1 minus tan x end fraction d x.

ln open vertical bar sin x minus cos x close vertical bar plus C

negative ln open vertical bar cos x plus sin x close vertical bar plus C

negative ln open vertical bar cos x minus sin x close vertical bar plus C

ln open vertical bar cos x plus sin x close vertical bar plus C

Question 15: Evaluate integral subscript 1 superscript 2 fraction numerator 6 over denominator open vertical bar 3 x close vertical bar square root of 9 x squared minus 4 end root end fraction d x.

sin to the power of negative 1 end exponent 3 minus sin to the power of negative 1 end exponent 3 over 2

tan to the power of negative 1 end exponent 3 minus tan to the power of negative 1 end exponent 3 over 2

s e c to the power of negative 1 end exponent 3 minus s e c to the power of negative 1 end exponent 3 over 2

Question 16: Find the area between the curves f left parenthesis x right parenthesis equals 1 minus 2 x and g left parenthesis x right parenthesis equals negative x minus 1 over the interval open square brackets negative 4 comma space minus 1 close square brackets.

13.5 space u n i t s squared

21 space u n i t s squared

27 space u n i t s squared

28.5 space u n i t s squared

Question 17: If R denotes a region bounded above by the graph of a continuous function f left parenthesis x right parenthesis, below by the x-axis, and on the left and right by the lines x equals a and x equals b, respectively, then which of the following integrals gives the mass of the lamina with density rho?

m equals rho integral subscript a superscript b open square brackets f left parenthesis x right parenthesis close square brackets squared over 2 space d x

m equals rho integral subscript a superscript b x f left parenthesis x right parenthesis space d x

m equals rho integral subscript a superscript b f left parenthesis x right parenthesis space d x

m equals rho integral subscript a superscript b open square brackets f left parenthesis x right parenthesis close square brackets squared space d x

Question 18: Evaluate fraction numerator d over denominator d x end fraction cos h left parenthesis 2 x squared plus 1 right parenthesis.

4 x sin h left parenthesis 2 x squared plus 1 right parenthesis

2 x sin h left parenthesis 2 x squared plus 1 right parenthesis

x sin h left parenthesis 2 x squared plus 1 right parenthesis

left parenthesis 2 x squared plus 1 right parenthesis sin h left parenthesis 2 x squared plus 1 right parenthesis

Question 19: Evaluate integral fraction numerator negative 1 over denominator open vertical bar x close vertical bar square root of 1 plus begin display style x squared over 25 end style end root end fraction space d x.

c s c h to the power of negative 1 end exponent open vertical bar x close vertical bar plus C

1 fifth c s c h to the power of negative 1 end exponent open vertical bar x over 5 close vertical bar plus C

c s c h to the power of negative 1 end exponent open vertical bar x over 5 close vertical bar plus C

1 fifth c s c h to the power of negative 1 end exponent open vertical bar x close vertical bar plus C

Question 20: Find the vertical and horizontal asymptotes of f left parenthesis x right parenthesis equals x plus sin x.

x equals 0 comma space y equals 0

x equals 1, no vertical asymptote

x equals negative straight pi over 2 comma space y equals straight pi over 2

No vertical and horizontal aymptotes.