40. ∫uncosudu=unsinu−n∫un−1sinudu∫uncosudu=unsinu−n∫un−1sinudu 41. ∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucos...
Calculus Volume 1Table of Contents Preface 1. Functions and Graphs 1 Introduction 1.1 Review of Functions 1.2 Basic Classes of Functions 1.3 Trigonometric Functions 1.4 Inverse Functions 1.5 Exponential and Logarithmic Functions Chapter 1 Review Exercises 2. Limits 2 Introduction 2.1 A...
Calculus Volume 1 2. Limits Search for: 2.3 The Limit LawsLearning Objectives Recognize the basic limit laws. Use the limit laws to evaluate the limit of a function. Evaluate the limit of a function by factoring. Use the limit laws to evaluate the limit of a polynomial or rational funct...
Calculus Volume 1 5. Integration Search for: 5.1 Approximating AreasLearning Objectives Use sigma (summation) notation to calculate sums and powers of integers. Use the sum of rectangular areas to approximate the area under a curve. Use Riemann sums to approximate area. Archimedes was fascinated...
Calculus Volume 1 4. Applications of Derivatives Search for: 4.4 The Mean Value TheoremLearning Objectives Explain the meaning of Rolle’s theorem. Describe the significance of the Mean Value Theorem. State three important consequences of the Mean Value Theorem. The Mean Value Theorem is one of...
Calculus Volume 1 6. Applications of Integration Search for: 6.4 Arc Length of a Curve and Surface AreaLearning Objectives Determine the length of a curve, y=f(x),y=f(x), between two points. Determine the length of a curve, x=g(y),x=g(y), between two points. Find th...
Calculus Volume 1 3. Derivatives Search for: 3.3 Differentiation RulesLearning Objectives State the constant, constant multiple, and power rules. Apply the sum and difference rules to combine derivatives. Use the product rule for finding the derivative of a product of functions. Use the quotient...
26. The volume of a right circular cone of radius xx and height yy is given by V=13πx2yV=13πx2y. Suppose that the volume of the cone is 85πcm385πcm3. Find dydxdydx when x=4x=4 and y=16y=16. Show Solution For the following exercises, consider a closed rectangular box wi...
Calculus Volume 1 3. Derivatives Search for: 3.5 Derivatives of Trigonometric FunctionsLearning Objectives Find the derivatives of the sine and cosine function. Find the derivatives of the standard trigonometric functions. Calculate the higher-order derivatives of the sine and cosine.One...
Alternatively, we can create a new function by composing two functions. For example, given the functions f(x)=x2f(x)=x2 and g(x)=3x+1g(x)=3x+1, the composite function f∘gf∘g is defined such that(f∘g)(x)=f(g(x))=(g(x))2=(3x+1)2(f∘g)(x)=f(g(x))=(g...