교과서 질문
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²√(9 - x²) on (-3,3)
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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
4장, 문제 4.1.41
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = 1/5 t⁵ - a⁴t
검증된 단계별 안내1
To find the critical points of the function ƒ(t) = 1/5 t⁵ - a⁴t, we first need to find its derivative ƒ'(t). The derivative will help us identify where the slope of the tangent to the curve is zero or undefined.
Differentiate the function ƒ(t) = 1/5 t⁵ - a⁴t with respect to t. Using the power rule, the derivative of 1/5 t⁵ is (1/5) * 5 * t⁴ = t⁴. The derivative of -a⁴t is -a⁴. Therefore, ƒ'(t) = t⁴ - a⁴.
Set the derivative ƒ'(t) equal to zero to find the critical points: t⁴ - a⁴ = 0. This equation will help us find the values of t where the function has critical points.
Solve the equation t⁴ - a⁴ = 0 for t. This can be rewritten as t⁴ = a⁴. Taking the fourth root of both sides, we find t = ±a.
The critical points of the function are t = a and t = -a. These are the values of t where the derivative is zero, indicating potential maxima, minima, or points of inflection.
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Critical Points
Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one typically takes the derivative of the function and solves for the values of the variable that satisfy these conditions.
추천 영상:
04:50
Critical Points
Derivative
The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. It is a fundamental concept in calculus, providing insights into the function's behavior, such as increasing or decreasing intervals. For the given function, the derivative will be calculated to find the critical points.
추천 영상:
가이드 코스
05:44Derivatives
Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, ƒ(t) = 1/5 t⁵ - a⁴t, is a polynomial of degree five. Understanding the properties of polynomial functions, such as their continuity and differentiability, is crucial for analyzing their critical points.
추천 영상:
6:04Introduction to Polynomial Functions
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{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.
f'(x) = 3x + sinx; f(0) = 3
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교과서 질문
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/x√(36x² - 36))dx
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{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.
y = 8/(x² + 4) (Witch of Agnesi)
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교과서 질문
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (6/√(4 - 4x²))dx
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교과서 질문
{Use of Tech} Estimating roots The values of various roots can be approximated using Newton’s method. For example, to approximate the value of ³√10, we let x = ³√10 and cube both sides of the equation to obtain x³ = 10, or x³ - 10 = 0. Therefore, ³√10 is a root of p(x) = x³ - 10, which we can approximate by applying Newton’s method. Approximate each value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of x₀ and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.
r = 7¹/⁴
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