BuyFind

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919
BuyFind

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919

Solutions

Chapter
Section
Chapter 11.2, Problem 38E
Textbook Problem

Finding a Particular Solution In Exercises 31-38, use the initial condition to find the particular solution of the differential equation. See Example 4.

Differential Equation Initial Condition

38 d y d x = 2 x y   sin   x 2

y=1 when x=0

Expert Solution

Want to see the full answer?

Check out a sample textbook solution.

Want to see this answer and more?

Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes!*

*Response times vary by subject and question complexity. Median response time is 34 minutes and may be longer for new subjects.

Chapter 11 Solutions

Calculus: An Applied Approach (MindTap Course List)
Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 1-12, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Verifying Solutions In Exercises 112, verify the...Ch. 11.1 - Determining Solutions In Exercises 1316, determine...Ch. 11.1 - Determining Solutions In Exercises 1316, determine...Ch. 11.1 - Determining Solutions In Exercises 1316, determine...Ch. 11.1 - Determining Solutions In Exercises 1316, determine...Ch. 11.1 - Determining Solutions In Exercises 1720, determine...Ch. 11.1 - Determining Solutions In Exercises 1720, determine...Ch. 11.1 - Determining Solutions In Exercises 1720, determine...Ch. 11.1 - Determining Solutions In Exercises 1720, determine...Ch. 11.1 - Finding a Particular Solution In Exercises 21-24,...Ch. 11.1 - Finding a Particular Solution In Exercises 21-24,...Ch. 11.1 - Finding a Particular Solution In Exercises 21-24,...Ch. 11.1 - Finding a Particular Solution In Exercises 21-24,...Ch. 11.1 - Finding a Particular Solution In Exercises 2528,...Ch. 11.1 - Finding a Particular Solution In Exercises 2528,...Ch. 11.1 - Finding a Particular Solution In Exercises 2528,...Ch. 11.1 - Finding a Particular Solution In Exercises 2528,...Ch. 11.1 - Sketching Graphs of Solutions In Exercises 29 and...Ch. 11.1 - Sketching Graphs of Solutions In Exercises 29 and...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Finding the General Solution In Exercises 31-38,...Ch. 11.1 - Biology The limiting capacity of the habitat of a...Ch. 11.1 - Investment The rate of growth of an investment is...Ch. 11.1 - Safety Assume the rate of change per hour in the...Ch. 11.1 - HOW DO YOU SEE IT? The graph shows a solution of...Ch. 11.1 - Verifying a Solution Show that y=a+Cek(1b)t is a...Ch. 11.1 - Using a Solution The function y=Cekx is a solution...Ch. 11.1 - True or False? In Exercises 45 and 46, determine...Ch. 11.1 - True or False? In Exercises 45 and 46, determine...Ch. 11.2 - Checkpoint 1 Worked-out solution available at...Ch. 11.2 - Checkpoint 2 Worked-out solution available at...Ch. 11.2 - Checkpoint 3 Worked-out solution available at...Ch. 11.2 - Checkpoint 4 Worked-out solution available at...Ch. 11.2 - Checkpoint 5 Worked-out solution available at...Ch. 11.2 - Checkpoint 6 Worked-out solution available at...Ch. 11.2 - In Exercises 1-6, find the indefinite integral....Ch. 11.2 - In Exercises 1-6, find the indefinite integral....Ch. 11.2 - In Exercises 1-6, find the indefinite integral....Ch. 11.2 - In Exercises 1-6, find the indefinite integral....Ch. 11.2 - In Exercises 1-6, find the indefinite integral....Ch. 11.2 - In Exercises 1-6, find the indefinite integral....Ch. 11.2 - In Exercises 7-10, solve the equation for C or k....Ch. 11.2 - In Exercises 7-10, solve the equation for C or k....Ch. 11.2 - In Exercises 7-10, solve the equation for C or k....Ch. 11.2 - In Exercises 7-10, solve the equation for C or k....Ch. 11.2 - Separation of Variables In Exercises 1-6, decide...Ch. 11.2 - Separation of Variables In Exercises 1-6, decide...Ch. 11.2 - Separation of Variables In Exercises 1-6, decide...Ch. 11.2 - Separation of Variables In Exercises 1-6, decide...Ch. 11.2 - Separation of Variables In Exercises 1-6, decide...Ch. 11.2 - Separation of Variables In Exercises 1-6, decide...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential Equation In Exercises 7-26,...Ch. 11.2 - Solving a Differential equation In Exercises 2730,...Ch. 11.2 - Solving a Differential equation In Exercises 2730,...Ch. 11.2 - Solving a Differential equation In Exercises 2730,...Ch. 11.2 - Solving a Differential equation In Exercises 2730,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding a Particular Solution In Exercises 31-38,...Ch. 11.2 - Finding an equation In Exercises 39 and 40, find...Ch. 11.2 - Finding an equation In Exercises 39 and 40, find...Ch. 11.2 - Velocity In Exercises 41 and 42, solve the...Ch. 11.2 - Velocity In Exercises 41 and 42, solve the...Ch. 11.2 - Biology: Cell Growth The growth rate of a...Ch. 11.2 - Solve the differential equation. HOW DO YOU SEE...Ch. 11.2 - Radioactive Decay The rate of decomposition of...Ch. 11.2 - Radioactive Decay The rate of decomposition of...Ch. 11.2 - Solve the differential equation. Weight Gain A...Ch. 11.2 - In Exercises 1-4, verify the solution of the...Ch. 11.2 - In Exercises 1-4, verify the solution of the...Ch. 11.2 - In Exercises 1-4, verify the solution of the...Ch. 11.2 - In Exercises 1-4, verify the solution of the...Ch. 11.2 - In Exercises 5 and 6, verify that the general...Ch. 11.2 - In Exercises 5 and 6, verify that the general...Ch. 11.2 - In Exercises 7-10, use separation of variables to...Ch. 11.2 - In Exercises 7-10, use separation of variables to...Ch. 11.2 - In Exercises 7-10, use separation of variables to...Ch. 11.2 - In Exercises 7-10, use separation of variables to...Ch. 11.2 - In Exercises 11 and 12, (a) find the general...Ch. 11.2 - In Exercises 11 and 12, (a) find the general...Ch. 11.2 - In Exercises 13 and 14, use the initial condition...Ch. 11.2 - In Exercises 13 and 14, use the initial condition...Ch. 11.2 - Find an equation of the graph that passes through...Ch. 11.2 - Ignoring resistance, a sailboat starting from rest...Ch. 11.3 - Checkpoint 1 Worked-out solution available at...Ch. 11.3 - Checkpoint 2 Worked-out solution available at...Ch. 11.3 - Checkpoint 3 Worked-out solution available at...Ch. 11.3 - In Exercises 1-4, simplify the expression....Ch. 11.3 - In Exercises 1-4, simplify the expression....Ch. 11.3 - In Exercises 1-4, simplify the expression. elnx3Ch. 11.3 - In Exercises 1-4, simplify the expression. e2lnx+xCh. 11.3 - In Exercises 5-10, find the indefinite integral....Ch. 11.3 - In Exercises 5-10, find the indefinite integral....Ch. 11.3 - In Exercises 5-10, find the indefinite integral....Ch. 11.3 - In Exercises 5-10, find the indefinite integral....Ch. 11.3 - In Exercises 5-10, find the indefinite integral....Ch. 11.3 - In Exercises 5-10, find the indefinite integral....Ch. 11.3 - Writing in Standard Form In Exercises 1-6, write...Ch. 11.3 - Writing in Standard Form In Exercises 1-6, write...Ch. 11.3 - Writing in Standard Form In Exercises 1-6, write...Ch. 11.3 - Writing in Standard Form In Exercises 1-6, write...Ch. 11.3 - Writing in Standard Form In Exercises 1-6, write...Ch. 11.3 - Writing in Standard Form In Exercises 1-6, write...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Solving a Linear Differential Equation In...Ch. 11.3 - Using Two Methods In Exercises 19-22, solve for y...Ch. 11.3 - Using Two Methods In Exercises 19-22, solve for y...Ch. 11.3 - Using Two Methods In Exercises 19-22, solve for y...Ch. 11.3 - Using Two Methods In Exercises 19-22, solve for y...Ch. 11.3 - Matching In Exercises 23-26, match the...Ch. 11.3 - Matching In Exercises 23-26, match the...Ch. 11.3 - Matching In Exercises 23-26, match the...Ch. 11.3 - Matching In Exercises 23-26, match the...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Finding a Particular Solution In Exercises 27-34,...Ch. 11.3 - Sales The rate of change (in thousands of units)...Ch. 11.3 - HOW DO YOU SEE IT? The rate of change in the...Ch. 11.3 - Vertical Motion A falling object encounters air...Ch. 11.3 - Velocity A booster rocket carrying an observation...Ch. 11.3 - Learning Curve The management at a medical supply...Ch. 11.3 - Investment Let A he the amount in a fund earning...Ch. 11.4 - Checkpoint 1 Worked out solution available at...Ch. 11.4 - Checkpoint2 Worked-out solution available at...Ch. 11.4 - Checkpoint 3 Worked-out solution available at...Ch. 11.4 - Checkpoint 4 Worked-out solution available at...Ch. 11.4 - Checkpoint 5 Worked-out solution available at...Ch. 11.4 - In Exercises 1-4, use separation of variables to...Ch. 11.4 - In Exercises 1-4, use separation of variables to...Ch. 11.4 - In Exercises 1-4, use separation of variables to...Ch. 11.4 - In Exercises 1-4, use separation of variables to...Ch. 11.4 - In Exercises 5-8, use an integrating factor to...Ch. 11.4 - In Exercises 5-8, use an integrating factor to...Ch. 11.4 - In Exercises 5-8, use an integrating factor to...Ch. 11.4 - In Exercises 5-8, use an integrating factor to...Ch. 11.4 - In Exercises 9 and 10, solve for k. 300=150e4kCh. 11.4 - In Exercises 9 and 10, solve for k. 0.36=1e7kCh. 11.4 - Chemical Reaction In Exercises 1 and 2, use the...Ch. 11.4 - Chemical Reaction In Exercises 1 and 2, use the...Ch. 11.4 - Advertising Awareness In Exercises 3 and 4, use...Ch. 11.4 - Advertising Awareness In Exercises 3 and 4, use...Ch. 11.4 - Population Growth In Exercises 5 and 6, use the...Ch. 11.4 - Population Growth In Exercises 5 and 6, use the...Ch. 11.4 - Hybrid Selection In Exercises 7 and 8, use the...Ch. 11.4 - Hybrid Selection In Exercises 7 and 8, use the...Ch. 11.4 - Finding a Particular Solution In Exercises 9-14,...Ch. 11.4 - Finding a Particular Solution In Exercises 9-14,...Ch. 11.4 - Finding a Particular Solution In Exercises 9-14,...Ch. 11.4 - Finding a Particular Solution In Exercises 9-14,...Ch. 11.4 - Finding a Particular Solution In Exercises 9-14,...Ch. 11.4 - Finding a Particular Solution In Exercises 914,...Ch. 11.4 - Chemical Reaction During a chemical reaction, a...Ch. 11.4 - Chemical Reaction Using the result of Exercise 15,...Ch. 11.4 - Population Growth The rate of change of the...Ch. 11.4 - Chemistry A wet towel hung from a clothesline to...Ch. 11.4 - Sales Growth The rate of change in sales S (in...Ch. 11.4 - Sales Growth Use the result of Exercise 19 to find...Ch. 11.4 - Biology A population of eight beavers has been...Ch. 11.4 - Biology A population of 30 rabbits has been...Ch. 11.4 - Hybrid Selection You are studying a population of...Ch. 11.4 - Hybrid Selection A research team is studying a...Ch. 11.4 - Chemical Mixture A 100-gallon tank is full of a...Ch. 11.4 - Chemical Mixture A 200-gallon tank is half full of...Ch. 11.4 - Population Growth When predicting population...Ch. 11.4 - HOW DO YOU SEE IT? In a learning theory project,...Ch. 11.4 - Investment A large corporation starts at time t=0...Ch. 11.4 - Investment In Exercises 32-34, use the result of...Ch. 11.4 - Investment In Exercises 32-34, use the result of...Ch. 11.4 - Investment In Exercises 32-34, use the result of...Ch. 11.4 - Medical Science In Exercises 35-38, a medical...Ch. 11.4 - Medical Science In Exercises 35-38, a medical...Ch. 11.4 - Medical Science In Exercises 35-38, a medical...Ch. 11.4 - Medical Science In Exercises 35-38, a medical...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Verifying Solutions In Exercises 1-10, verify the...Ch. 11 - Determining Solutions In Exercises 11-14,...Ch. 11 - Determining Solutions In Exercises 11-14,...Ch. 11 - Determining Solutions In Exercises 11-14,...Ch. 11 - Determining Solutions In Exercises 11-14,...Ch. 11 - Finding a Particular Solution In Exercises 15 and...Ch. 11 - Finding a Particular Solution In Exercises 15 and...Ch. 11 - Finding a Particular Solution In Exercises 17 and...Ch. 11 - Finding a Particular Solution In Exercises 17 and...Ch. 11 - Sketching Graphs of Solutions In Exercises 19 and...Ch. 11 - Sketching Graphs of Solutions In Exercises 19 and...Ch. 11 - Finding the General Solution In Exercises 21-26,...Ch. 11 - Finding the General Solution In Exercises 21-26,...Ch. 11 - Finding the General Solution In Exercises 21-26,...Ch. 11 - Finding the General Solution In Exercises 21-26,...Ch. 11 - Finding the General Solution In Exercises 21-26,...Ch. 11 - Finding the General Solution In Exercises 21-26,...Ch. 11 - Separation of Variables In Exercises 27-30, decide...Ch. 11 - Separation of Variables In Exercises 27-30, decide...Ch. 11 - Separation of Variables In Exercises 27-30, decide...Ch. 11 - Separation of Variables In Exercises 27-30, decide...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises...Ch. 11 - Solving a Differential Equation In Exercises 43...Ch. 11 - Solving a Differential Equation In Exercises 43...Ch. 11 - Finding a Particular Solution In Exercises 45-48,...Ch. 11 - Finding a Particular Solution In Exercises 45-48,...Ch. 11 - Finding a Particular Solution In Exercises 4548,...Ch. 11 - Finding a Particular Solution In Exercises 45-48,...Ch. 11 - Finding an Equation In Exercises 49 and 50, find...Ch. 11 - Finding an Equation In Exercises 49 and 50, find...Ch. 11 - Radioactive Decay The rate of decomposition of...Ch. 11 - Advertising Awareness A newly opened chiropractors...Ch. 11 - Writing in Standard Form In Exercises 53-56, write...Ch. 11 - Writing in Standard Form In Exercises 53-56, write...Ch. 11 - Writing in Standard Form In Exercises 53-56, write...Ch. 11 - Writing in Standard Form In Exercises 53-56, write...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Solving a Linear Differential Equation In...Ch. 11 - Finding a Particular Solution In Exercises 65 and...Ch. 11 - Finding a Particular Solution In Exercises 65 and...Ch. 11 - Safety Assume the rate of change per hour in the...Ch. 11 - Chemistry A wet shirt hung from a clothesline to...Ch. 11 - Economics: Paretos Law According to the economist...Ch. 11 - Economics: Paretos Law In 2013, 29.7 million...Ch. 11 - Chemical Reaction During a chemical reaction, a...Ch. 11 - Chemical Reaction Using the result of Exercise 71,...Ch. 11 - Chemistry: Newtons Law of Cooling In Exercises 73...Ch. 11 - Chemistry: Newtons Law of Cooling In Exercises 73...Ch. 11 - Biology A population of 12 pelicans has been...Ch. 11 - Chemical Mixture A tank contains 30 gallons of a...Ch. 11 - Chemical Mixture A tank contains 20 gallons of a...Ch. 11 - In Exercises 1 and 2, verify the solution of the...Ch. 11 - In Exercises 1 and 2, verify the solution of the...Ch. 11 - In Exercises 3-5, use separation of variables to...Ch. 11 - In Exercises 3-5, use separation of variables to...Ch. 11 - In Exercises 3-5, use separation of variables to...Ch. 11 - In Exercises 6-8, find the general solution of the...Ch. 11 - In Exercises 6-8, find the general solution of the...Ch. 11 - In Exercises 6-8, find the general solution of the...Ch. 11 - In Exercises 9-11, use the initial condition to...Ch. 11 - In Exercises 9-11, use the initial condition to...Ch. 11 - In Exercises 9-11, use the initial condition to...Ch. 11 - A lamb that weighs 10 pounds at birth gains weight...Ch. 11 - An infectious disease spreads through a large...

Additional Math Textbook Solutions

Find more solutions based on key concepts
(a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5. Theorem 4

Single Variable Calculus: Early Transcendentals, Volume I

Sketch the graphs of the equations in Exercises 512. xy=4

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the mean, median, and mode for the following scores; 8, 7, 5, 7, 0, 10, 2, 4, 11, 7, 8, 7

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Fill in the blanks. 6. If L1 and L2 are nonvertical lines with slopes m1 and m2, respectively, then L1 is paral...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Prove each identity. csc+sin()=cos2sin

Trigonometry (MindTap Course List)

The first step to evaluate cos4 x dx is to rewrite the integral as a) (1+cos2x2)2dx b) cos3xcosxdx c) (1sin4x)...

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False:

Study Guide for Stewart's Multivariable Calculus, 8th

Answer Problems 112 without referring back to the text. Fill in the blanks or answer true or false. 4. The line...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

The following 20 observations are for two quantitative variables, x and y. Develop a scatter diagram for the r...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Reminder Round all answers to decimal places unless otherwise indicated. Mileage for an Old Car The gas mileage...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)