Textbook Question
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = 2 + 3 sin t; v(0) = 1, s(0) = 10
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 4.1.45
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x³ - 3x² on [-1, 3]
Verified step by step guidance1
First, understand that absolute extrema refer to the highest and lowest values of a function on a given interval. To find these, we need to evaluate the function at critical points and endpoints of the interval.
Find the derivative of the function ƒ(x) = x³ - 3x². The derivative, ƒ'(x), will help us identify critical points where the slope is zero or undefined. Calculate ƒ'(x) = 3x² - 6x.
Set the derivative ƒ'(x) = 0 to find critical points. Solve the equation 3x² - 6x = 0. Factor the equation to get 3x(x - 2) = 0, which gives the critical points x = 0 and x = 2.
Evaluate the function ƒ(x) at the critical points and at the endpoints of the interval [-1, 3]. Calculate ƒ(-1), ƒ(0), ƒ(2), and ƒ(3) to find the values of the function at these points.
Compare the values obtained from evaluating the function at the critical points and endpoints. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum on the interval [-1, 3].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Points
Critical points are values of x in the domain of a function where the derivative is either zero or undefined. These points are essential for finding local maxima and minima, as they indicate where the function's slope changes. To locate absolute extrema, one must first find these critical points within the given interval.
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Critical Points
Endpoints of the Interval
When determining absolute extrema on a closed interval, it is crucial to evaluate the function at both the critical points and the endpoints of the interval. The absolute maximum or minimum could occur at these endpoints, so they must be included in the analysis to ensure all potential extreme values are considered.
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05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
First Derivative Test
The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function around that point. This test helps in identifying the nature of the extrema found in the analysis.
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07:09The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question
Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.
f"(x) > 0 on (-∞,-2); f"(-2) = 0; f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4,∞)
Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4/x√(x² - 1))dx
Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = √2 sin x- x on [0, 2π]
Textbook Question
{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.
y = 4√x and y = x² + 1
Textbook Question
Linear approximation Find the linear approximation to the following functions at the given point a.
g(t) = √(2t + 9); a = -4