Textbook Question
Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels.
a. How many barrels are produced in the first 35 days?
Ch. 6 - Applications of Integration
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 6, Problem 6.1.21a
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 9−t² on [0, 4]; s(0)=−2
Verified step by step guidance1
Recall that the position function \(s(t)\) is the antiderivative (integral) of the velocity function \(v(t)\) plus the initial position constant. Mathematically, this is expressed as \(s(t) = \int v(t) \, dt + C\).
Given the velocity function \(v(t) = 9 - t^{2}\), set up the integral to find the position function: \(s(t) = \int (9 - t^{2}) \, dt + C\).
Integrate each term separately: the integral of 9 with respect to \(t\) is \$9t\(, and the integral of \)-t^{2}\( with respect to \)t$ is \(-\frac{t^{3}}{3}\). So, \(s(t) = 9t - \frac{t^{3}}{3} + C\).
Use the initial condition \(s(0) = -2\) to solve for the constant \(C\). Substitute \(t=0\) into the position function: \(s(0) = 9 \cdot 0 - \frac{0^{3}}{3} + C = C\). Since \(s(0) = -2\), it follows that \(C = -2\).
Write the final position function incorporating the constant: \(s(t) = 9t - \frac{t^{3}}{3} - 2\) for \(t \geq 0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Position Relationship
Velocity is the rate of change of position with respect to time. The position function s(t) describes the location of an object at time t, and its derivative s'(t) equals the velocity v(t). Understanding this relationship allows us to find position by integrating velocity.
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Derivatives Applied To Velocity
Antiderivative (Indefinite Integral)
The antiderivative of a function is another function whose derivative is the original function. To find position from velocity, we compute the antiderivative of v(t), which gives s(t) plus a constant of integration. This constant is determined using initial conditions.
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05:04Introduction to Indefinite Integrals
Initial Conditions and Constants of Integration
When integrating velocity to find position, an unknown constant appears. The initial condition, such as s(0) = -2, allows us to solve for this constant, ensuring the position function accurately reflects the object's starting location.
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05:03Initial Value Problems
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The distance traveled by an object moving along a line is the same as the displacement of the object.
Textbook Question
{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0.
a. Determine the position function, for t≥0.
Textbook Question
For the given regions R₁ and R₂, complete the following steps.
a. Find the area of region R₁.
R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.
Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
Textbook Question
A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.
a. Using calculus