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College Calculus: Level I Related Rates

Section 4: Derivatives, part 2: Lecture 5 | 29:05 min

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Post by Alan Delez on October 4, 2012

A bug is moving along the right side of the parabola y=x^2 at a rate such that its distance from the origin is increasing at 1cm/min. At what rates are the x- and y-coordinates of the bug increasing when the bug is at the point (2,4)?

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Post by Rebecca Kerns on May 5, 2010

example 4 does not finish, it just starts over when the instructor get to the rate of water.

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Related Rates

You will be differentiating both sides of an equation with respect to .
Many problems here are geometrical – I strongly advise drawing a sketch!
Use variables for those quantities that vary with other quantities in the problem.
Wait to substitute in specific variable values until after you have differentiated!

Related Rates

The radius of a sphere is growing at a rate of 1 [m/s]. Find the rate at which the area is changing.

A = πr²
[d/dt]A = [d/dt]πr²
[dA/dt] = π(2r) [dr/dt]

[dA/dt] = 2 πr (1 [m/s])

The radius of a sphere is growing at a rate of 1 [m/s]. Find the rate at which the circumference is changing.

C = 2 πr
[d/dt]C = [d/dt]2 πr
[dC/dt] = 2 π[dr/dt]
[dC/dt] = 2 π[dr/dt] 1 [m/s]

[dC/dt] = 2 π[m/s]

A sphere's volume is decreasing at a rate of 10 [(m³)/s] from an initial volume V₀ = 100 m³. How big is the diameter after 5 seconds of decreasing?

How does the volume relate to the diameter?
Volume of a sphere is [4/3] πr³
V = [4/3] π([d/2])³
Find the volume after 5 seconds. To find the current volume, we need to add the initial volume and the rate of change multiplied by time.
V = V₀ + [dV/dt] t
V = 100 m³ − 10 [(m³)/s] 5 s = 50 m³
We subtracted above because the problem said the volume was decreasing. Now we plug in to the volume formula and solve for d.
50 m³ = [4/3] π([d/2])³
[(150 m³)/(4 π)] = ([d/2])³
[d/2] = ³√{[(150 m³)/(4 π)]}

d = 2 ³√{[150/(4 π)]} m

A hose is pouring water at a rate of 3 [(m³)/s] into a rigid cylindrical container. The container has a radius of 10 m. What is the rate at which the height of the water level rises? (Volume of a cylinder = πh r²)

We need a formula that relates the volume of a cylinder and its height.
V = πh r²
The problem says its a rigid container, so we can treat the radius as a constant. Otherwise, we would have to use the product rule when taking the derivative.
[d/dt]V = [d/dt]πh r²
[dV/dt] = πr² [dh/dt]
[dh/dt] = [dV/dt] [1/(πr²)]
[dh/dt] = 3 [(m³)/s] [1/(π(10 m)²)]

[dh/dt] = [3/(100π)] [m/s]

A locomotive's position is given by x(t) = 36 [m/s] t. A train car unhitches from the engine and begins to slow. Its position is given by x(t) = −20 [m/s] t. At what rate is the distance between the two trains growing?

In this case, the trains are traveling along the same axis, so all we have to do here is add the speeds.
v(t) = [d/dt]36 [m/s] t + [d/dt](−20 [m/s] t)

v(t) = 16 [m/s]

A person's Body Mass Index in imperial units is given by BMI = [(703 weight)/(height²)]. A 70 inch person is increasing their BMI by 1 [lb/(in²)] every year while maintaining a constant height. At what rate is their weight changing?

[d/dt]BMI = [d/dt][(703 weight)/(height²)]
[d/dt]BMI = [703/(height²)] [d/dt]weight
[d/dt]weight = [d/dt]BMI [(height²)/703]
[d/dt]weight = 1 [lb/(in² year)] [((70 in)²)/703]
[d/dt]weight = [4900/703] [lb/year]

The person's weight is increasing at a rate of about 7 lbs per year

The pressure of a gas in a container is given by P = [nRT/V] where n is the number of molecules, R is a constant, T is the temperature, and V is the volume. Given that the volume is constant, solve for the rate of change of pressure.

[d/dt]P = [d/dt][nRT/V]
[dP/dt] = [R/V] [d/dt]nT
[dP/dt] = [R/V] (n [d/dt]T + T [d/dt]n)
[dP/dt] = [R/V] (n [dT/dt] + T [dn/dt])

[dP/dt] = [R/V] (n [dT/dt] + T [dn/dt])

Using the pressure formula of the previous problem, find the rate of change of P with respect to time given that n and T are constants.

[d/dt]P = [d/dt][nRT/V]
[dP/dt] = nRT [d/dt][1/V]
We can use the quotient rule, or treat it as a negative exponent.
[dP/dt] = nRT [d/dt]V⁻¹
[dP/dt] = nRT (−V⁻²) [d/dt]V

[dP/dt] = −[nRT/(V²)] [dV/dt]

Water pressure at a depth is given by P = P₀ + ρg h. At the surface of water, P₀ = 10⁵ [kg/(s² m)], ρ = 1000 [kg/(m³)], g = 9.8 [m/(s²)], and h is the height or depth of the water. If pressure is increasing at a rate of 9800 [kg/(s³ m)], how fast are you descending?

[d/dt]P = [d/dt]P₀ + [d/dt]ρg h
[dP/dt] = ρg [dh/dt]
[dh/dt] = [([dP/dt])/(ρg)]
[dh/dt] = [(9800 [kg/(s³ m)])/(1000 [kg/(m³)] 9.8 [m/(s²)])]

[dh/dt] = 1 [m/s]

A cube initially has sides 1 m in length. The length of the sides is increasing at 2 [m/s]. At what rate is the volume increasing when the length is 4 m?

V = l³
[d/dt]V = [d/dt]l³
[dV/dt] = 3 l² [d/dt]l
[dV/dt] = 3 (4 m)² 2 [m/s]

[dV/dt] = 96 [(m³)/s]

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Related Rates

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

What Are Related Rates?

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

Intro 0:00
What Are Related Rates? 0:08
Lecture Example 1 0:35
Lecture Example 2 5:25
Lecture Example 3 11:54
Additional Example 4
Additional Example 5

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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Related Rates

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro