Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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63,186 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This module provides an overview of several key mathematical concepts including angle measurements in radians, methods for handling radicals and rational expressions, various techniques for integration such as substitution and Riemann sums, and the analysis of rates of change through derivatives and related rates. In addition, it covers fundamental statistical concepts like random variables, expected value, and variance, highlighting both theoretical approaches and practical applications in fields like economics and scientific measuring techniques.

Learning Objectives

1

Convert angles between degrees and radians and understand their significance in trigonometry.

2

Apply techniques for simplifying and differentiating radicals and rational expressions.

3

Solve integrals using substitution and Riemann sum approximations for definite integrals.

4

Analyze rates of change through derivatives, related rates, and applications in marginal revenue and population models.

5

Understand and compute key statistical measures such as expected value and variance for various probability distributions.

Key Concepts

CONCEPT

DEFINITION

Radian

A unit of angular measure where one radian is the angle created when the arc length equals the radius of the circle. Conversion formula: Radians = Degrees × (π/180).

Radical

An expression that includes a root, such as a square root or cube root. Differentiating and simplifying radicals are common algebraic tasks.

Integration by Substitution

A method used to evaluate integrals and solve problems by substituting part of the integrand with a new variable to simplify the integration process.

Riemann Sum

A method for approximating the definite integral by summing the areas of a sequence of rectangles that approximate the area under a curve.

Random Variable

A variable whose value is subject to variations due to chance, and can be discrete or continuous, often characterized by probability distributions.

Expected Value

The weighted average of all possible values of a random variable, computed by multiplying each outcome by its probability and summing the results.

Derivative

A measure of the instantaneous rate of change of a function with respect to one of its variables, often interpreted as the slope of the tangent line.

Related Rates

Problems that involve finding the rate at which one quantity changes with respect to another when the quantities are related through an equation.

Example Problems

Example 1

Determine the growth constant $k$, then find all solutions of the given differential equation. $$y^{\prime}=y$$

Example 2

Determine the growth constant $k$, then find all solutions of the given differential equation. $$y^{\prime}=.4 y$$

Example 3

Determine the growth constant $k$, then find all solutions of the given differential equation. $$y^{\prime}=1.7 y$$

Example 4

Determine the growth constant $k$, then find all solutions of the given differential equation. $$y^{\prime}=\frac{y}{4}$$

Example 5

Determine the growth constant $k$, then find all solutions of the given differential equation. $$y^{\prime}-\frac{y}{2}=0$$
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Step-by-Step Explanations

QUESTION

Convert 394° to radians.

STEP-BY-STEP ANSWER:

Step 1: Write down the conversion factor: π radians = 180°.
Step 2: Multiply 394 by (π/180): 394 × (π/180).
Step 3: Simplify the expression if possible (394 and 180 can sometimes be reduced): This gives (394Ï€)/180 which can also be written as (197Ï€)/90.
Final Answer: 394° = (197π)/90 radians.

Degree to Radian Conversion

QUESTION

Evaluate the integral ∫2x cos(x²) dx using substitution.

STEP-BY-STEP ANSWER:

Step 1: Identify the substitution. Let u = x², then differentiate to get du = 2x dx.
Step 2: Replace 2x dx in the integral with du. The integral becomes ∫cos(u) du.
Step 3: Integrate cos(u) with respect to u to obtain sin(u) + C.
Step 4: Substitute back u = x² to get sin(x²) + C.
Final Answer: ∫2x cos(x²) dx = sin(x²) + C.

Integration by Substitution

QUESTION

Calculate the expected value for a random variable X with outcomes 1, 2, 3 and probabilities 0.2, 0.5, 0.3 respectively.

STEP-BY-STEP ANSWER:

Step 1: Multiply each outcome by its corresponding probability: (1×0.2), (2×0.5), and (3×0.3).
Step 2: Compute these products: 0.2, 1.0, and 0.9.
Step 3: Sum the results: 0.2 + 1.0 + 0.9 = 2.1.
Final Answer: The expected value is 2.1.

Expected Value of a Discrete Random Variable

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Common Mistakes

  • Using the incorrect conversion factor when switching between degrees and radians.
  • Neglecting to change the integration limits or correctly substitute variables in integration by substitution.
  • Miscomputing weights in the expected value calculation by not matching outcomes with their corresponding probabilities.
  • Mixing up average rate of change and instantaneous rate of change when analyzing derivatives.
  • Overlooking conditions for convergence when working with infinite series or Riemann sum approximations.