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 6.7.59b
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?
Verified step by step guidance1
Recall that the work done by a variable force \( F(x) \) in moving an object from position \( a \) to \( b \) is given by the definite integral of the force over that interval: \[ W = \int_{a}^{b} F(x) \, dx \].
Identify the limits of integration from the problem: the spring is stretched from the equilibrium position \( x=0 \) to \( x=1.5 \), so \( a=0 \) and \( b=1.5 \).
Substitute the given force function \( F(x) = 16x - 0.1x^3 \) into the integral: \[ W = \int_{0}^{1.5} (16x - 0.1x^3) \, dx \].
Split the integral into two simpler integrals: \[ W = \int_{0}^{1.5} 16x \, dx - \int_{0}^{1.5} 0.1x^3 \, dx \].
Integrate each term using the power rule for integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]. So, \[ \int 16x \, dx = 16 \cdot \frac{x^2}{2} = 8x^2 \] and \[ \int 0.1x^3 \, dx = 0.1 \cdot \frac{x^4}{4} = 0.025x^4 \]. Then evaluate these expressions at the limits \( 0 \) and \( 1.5 \) and subtract to find the work done.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Work Done by a Variable Force
Work done by a force that varies with position is calculated by integrating the force function over the displacement interval. For a spring, this means integrating the restoring force from the initial to the final position to find the total work required to stretch or compress it.
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05:40
Work Done On A Spring (Hooke's Law)
Nonlinear Spring Force
Unlike Hooke’s law for linear springs (F = kx), a nonlinear spring has a restoring force that depends on higher powers of displacement, such as F(x) = 16x − 0.1x³. This means the force changes in a more complex way as the spring stretches, affecting the work calculation.
Recommended video:
05:40Work Done On A Spring (Hooke's Law)
Definite Integral in Calculus
A definite integral computes the accumulation of quantities, such as work done, over an interval. In this problem, integrating the force function from 0 to 1.5 gives the total work done in stretching the spring, capturing the effect of the nonlinear force.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.
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Textbook Question
Consider the following curves on the given intervals.
b. Use a calculator or software to approximate the surface area.
y=tan x , for 0≤x≤π/4; about the x-axis
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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
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.
b. The distance traveled between t=0 and t=5
Textbook Question
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.