Textbook Question
71. Different Methods
Let I = ∫ (x²)/(x + 1) dx.
b. Evaluate I by first performing long division on the integrand.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.8.66b
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
b. Calculate f''(x).
Verified step by step guidance1
Recall that the function is given by \(f(x) = \cos(x^2)\). Our goal is to find the second derivative \(f''(x)\).
First, find the first derivative \(f'(x)\) by applying the chain rule. The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\), and here \(u = x^2\). So, \(f'(x) = -\sin(x^2) \cdot \frac{d}{dx}(x^2)\).
Calculate the derivative of the inner function: \(\frac{d}{dx}(x^2) = 2x\). Substitute this back to get \(f'(x) = -2x \sin(x^2)\).
Next, find the second derivative \(f''(x)\) by differentiating \(f'(x) = -2x \sin(x^2)\). Use the product rule: if \(h(x) = u(x)v(x)\), then \(h'(x) = u'(x)v(x) + u(x)v'(x)\).
Let \(u(x) = -2x\) and \(v(x) = \sin(x^2)\). Then, \(u'(x) = -2\) and \(v'(x)\) requires the chain rule again: \(v'(x) = \cos(x^2) \cdot 2x\). Substitute these into the product rule formula to express \(f''(x)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second Derivative
The second derivative of a function measures the rate of change of the first derivative, providing information about the function's concavity and acceleration. It is found by differentiating the first derivative once more with respect to the variable.
Recommended video:
06:02
The Second Derivative Test: Finding Local Extrema
Chain Rule
The chain rule is a differentiation technique used when dealing with composite functions. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x), allowing us to differentiate functions like cos(x²) effectively.
Recommended video:
05:02Intro to the Chain Rule
Derivative of Trigonometric Functions
Understanding the derivatives of basic trigonometric functions, such as d/dx[cos(x)] = -sin(x), is essential. This knowledge helps in differentiating more complex functions involving trigonometric expressions.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
b. ∫ 7x e³ˣ dx
Textbook Question
A piece of wood paneling must be cut in the shape shown in the figure.
The coordinates of several points on its curved surface are also shown (with units of inches).
a. Estimate the surface area of the paneling using the Trapezoid Rule.
Textbook Question
93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
b. Which car travels farthest on the interval 0 ≤ t ≤ 5?
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.