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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.9.93
Chapter 4, Problem 4.9.93

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.


v(t) = 2√t; s(0) = 1

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Step 1: Recall that the position function s(t) can be found by integrating the velocity function v(t) with respect to time t. The relationship is given by: s(t) = ∫v(t) dt + C, where C is the constant of integration.
Step 2: Substitute the given velocity function v(t) = 2√t into the integral. This gives: s(t) = ∫2√t dt + C.
Step 3: Simplify the integral. Recall that √t can be written as t^{1/2}. Using the power rule for integration, ∫t^n dt = (t^{n+1})/(n+1), apply this to integrate 2t^{1/2}.
Step 4: After integrating, you will have an expression for s(t) in terms of t and the constant of integration C. Use the initial condition s(0) = 1 to solve for C. Substitute t = 0 into the position function and set s(0) equal to 1.
Step 5: Combine the result of the integration and the value of C to write the final position function s(t).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity and Position Functions

Velocity is the rate of change of position with respect to time, represented mathematically as the derivative of the position function. To find the position function from a given velocity function, one must integrate the velocity function. This relationship is fundamental in calculus, as it connects the concepts of motion and change.
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Integration

Integration is the process of finding the integral of a function, which essentially reverses differentiation. In the context of finding a position function from a velocity function, integration allows us to determine the accumulated position over time. The result of integration includes a constant of integration, which can be determined using initial conditions.
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Initial Conditions

Initial conditions are specific values that help define a particular solution to a differential equation or integral. In this case, the initial position s(0) = 1 provides the necessary information to solve for the constant of integration after finding the position function. This ensures that the solution accurately reflects the scenario described in the problem.
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Related Practice
Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π/2⁻ (π - 2x) tan x  

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Textbook Question

Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of 16 ft³. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?

Textbook Question

{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.


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Textbook Question

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ƒ(x) = x³ -4a²x

Textbook Question

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ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]

Textbook Question

Rectangles beneath a semicircle A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?