Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((1 + √x)/x)dx
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Ch. 4 - Applications of the Derivative
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 4, Problem 4.6.62
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = sin² x
Verified step by step guidance1
First, identify the function given: \( f(x) = \sin^2 x \). This is the function whose differential we need to find.
To find \( dy \), we need to determine \( f'(x) \), the derivative of \( f(x) \). Use the chain rule to differentiate \( \sin^2 x \). Let \( u = \sin x \), then \( f(x) = u^2 \).
Differentiate \( u^2 \) with respect to \( u \) to get \( 2u \). Then differentiate \( \sin x \) with respect to \( x \) to get \( \cos x \).
Apply the chain rule: \( f'(x) = 2u \cdot \cos x = 2 \sin x \cdot \cos x \).
Express the differential relationship: \( dy = f'(x)dx = 2 \sin x \cos x \cdot dx \). This shows how a small change in \( x \) results in a change in \( y \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentials
Differentials represent the infinitesimal changes in variables. In calculus, if y is a function of x, the differential dy is defined as the product of the derivative f'(x) and the differential dx, which represents a small change in x. This relationship helps in approximating how a small change in x affects the change in y.
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Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is denoted as f'(x) and is calculated as the limit of the average rate of change of the function as the interval approaches zero. In the context of the given function, f(x) = sin² x, the derivative will provide the slope of the tangent line at any point on the curve.
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05:44Derivatives
Chain Rule
The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This is particularly useful for functions like f(x) = sin² x, where the inner function is sin x.
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Related Practice
Textbook Question
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = cos x
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Textbook Question
Sketch a continuous function f on some interval that has the properties described. Answers will vary.
The function f satisfies f'(-2) = 2, f'(0) = 0, f'(1) = -3 and f'(4) = 1.
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Textbook Question
Suppose f is differentiable on (-∞,∞) and f(5.01) - f(5) = 0.25.Use linear approximation to estimate the value of f'(5).
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Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→1⁻ (1-x) tan πx/2
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Textbook Question
Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?
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