Textbook Question
55–58. Marginal cost Consider the following marginal cost functions.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
C′(x)=200−0.05x
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 6.1.50b
Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.
b. Find the function that gives the total blood pumped between t=0 and a future time t>0.
Verified step by step guidance1
Identify the given rate of blood flow as the derivative of the volume function: \(V'(t) = 70(1 + \sin(2\pi t))\) milliliters per second.
Recall that to find the total volume pumped over time, you need to find the original function \(V(t)\) by integrating the rate function \(V'(t)\) with respect to \(t\).
Set up the integral: \(V(t) = \int 70(1 + \sin(2\pi t)) \, dt\).
Split the integral into two parts for easier integration: \(V(t) = 70 \int 1 \, dt + 70 \int \sin(2\pi t) \, dt\).
Integrate each part separately, remembering to include the constant of integration \(C\), and then use the initial condition \(V(0) = 0\) to solve for \(C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Understanding the Derivative as a Rate of Change
The derivative V′(t) represents the instantaneous rate at which blood is pumped by the heart at time t. Recognizing that V′(t) is the rate of change of the total volume V(t) is essential to relate the given function to the total blood pumped over time.
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Intro To Related Rates
Definite Integration to Find Accumulated Quantity
To find the total blood pumped from time 0 to t, we integrate the rate function V′(t) over [0, t]. Integration accumulates the instantaneous rates over time, yielding the total volume V(t). This process reverses differentiation and uses the initial condition to find the constant of integration.
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Applying Initial Conditions to Determine the Integration Constant
Since V(0) = 0, the initial amount of blood pumped is zero. After integrating V′(t), we use this initial condition to solve for the constant of integration, ensuring the function V(t) accurately models the total blood volume pumped starting from zero.
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Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. If a region is revolved about the y-axis, then the shell method must be used.
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Textbook Question
A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7.
b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?
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Textbook Question
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
b. Find the displacement over the given interval.
v(t) = 3t²−6t on [0, 3]
Textbook Question
40–43. Population growth
When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.
b. Find the population P(t) at any time t≥0.
Textbook Question
Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.
b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.