Textbook Question
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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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 4.9.97
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) = -32; v(0) = 20, s(0) = 0
Verified step by step guidance1
Start by recalling the relationship between acceleration, velocity, and position. Acceleration is the derivative of velocity, and velocity is the derivative of position. To find the position function, we need to integrate the acceleration function twice.
Integrate the acceleration function a(t) = -32 with respect to time t to find the velocity function v(t). This gives v(t) = ∫(-32) dt = -32t + C₁, where C₁ is the constant of integration.
Use the initial condition v(0) = 20 to solve for C₁. Substitute t = 0 and v(0) = 20 into the velocity equation: 20 = -32(0) + C₁. Solve for C₁.
Now integrate the velocity function v(t) = -32t + C₁ with respect to time t to find the position function s(t). This gives s(t) = ∫(-32t + C₁) dt = -16t² + C₁t + C₂, where C₂ is another constant of integration.
Use the initial condition s(0) = 0 to solve for C₂. Substitute t = 0 and s(0) = 0 into the position equation: 0 = -16(0)² + C₁(0) + C₂. Solve for C₂. Once C₁ and C₂ are determined, substitute them back into the position function s(t) to get the final expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Acceleration
Acceleration is the rate of change of velocity with respect to time. In this context, the acceleration function a(t) = -32 indicates a constant acceleration, which means the object's velocity decreases uniformly over time. Understanding acceleration is crucial for determining how it affects the object's velocity and position as it moves along a line.
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Derivatives Applied To Acceleration
Velocity
Velocity is the integral of acceleration and represents the rate of change of position with respect to time. Given the initial velocity v(0) = 20, we can find the velocity function by integrating the acceleration function. This step is essential for determining how the object's speed and direction change over time, which directly influences its position.
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Position Function
The position function describes the location of an object at any given time and is obtained by integrating the velocity function. With the initial position s(0) = 0, we can establish the position function by integrating the velocity function derived from the acceleration. This concept is vital for understanding the object's trajectory and final position after a certain time.
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Related Practice
Textbook Question
{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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Textbook Question
{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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Textbook Question
Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 - x². What are the dimensions of the rectangle with the maximum area? What is the area?
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Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = t/ t² + 1
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Textbook Question
A graph of ƒ and the lines tangent to ƒ at x = 1, 2 and 3 are given. If x₀ = 3, find the values of x₁, x₂, and x₃, that are obtained by applying Newton’s method. <IMAGE>
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