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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.95c
Chapter 3, Problem 3.95c

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.


c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Verified step by step guidance
1
Start by identifying the given equation for the total surface area of the cylinder: \( S = 2\pi r^2 + 2\pi rh \).
Recognize that \( S \), \( r \), and \( h \) are all functions of time \( t \), so we need to use implicit differentiation with respect to \( t \).
Differentiate both sides of the equation with respect to \( t \). For the first term \( 2\pi r^2 \), use the chain rule: \( \frac{d}{dt}(2\pi r^2) = 4\pi r \frac{dr}{dt} \).
For the second term \( 2\pi rh \), apply the product rule: \( \frac{d}{dt}(2\pi rh) = 2\pi \left( r \frac{dh}{dt} + h \frac{dr}{dt} \right) \).
Combine the differentiated terms to express \( \frac{dS}{dt} \) in terms of \( \frac{dr}{dt} \) and \( \frac{dh}{dt} \): \( \frac{dS}{dt} = 4\pi r \frac{dr}{dt} + 2\pi \left( r \frac{dh}{dt} + h \frac{dr}{dt} \right) \).

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Key Concepts

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

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this context, we are interested in how the surface area of a cylinder changes with respect to time as both the radius and height change. This concept is fundamental in calculus, as it allows us to connect different rates of change through derivatives.
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Intro To Related Rates

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. In the context of the given problem, it allows us to express the derivative of the surface area with respect to time as a function of the derivatives of the radius and height. This is crucial for relating the rates of change of different variables.
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Intro to the Chain Rule

Surface Area of a Cylinder

The surface area of a right circular cylinder is given by the formula S = 2πr² + 2πrh, which includes contributions from both the circular bases and the lateral surface. Understanding this formula is essential for applying related rates, as it provides the relationship between the radius, height, and surface area that we need to differentiate with respect to time.
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Example 1: Minimizing Surface Area
Related Practice
Textbook Question

Computer Explorations


Use a CAS to perform the following steps in Exercises 55–62.


b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.


2y² + (xy)¹/³ = x² + 2, P(1,1)

Textbook Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s


b. area?

Textbook Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


c. f(x) / (g(x) + 1), x = 1

Textbook Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


c. ƒ(x) , x = 1

g(x) + 1

Textbook Question

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

c. How are dx/dt, dy/dt, and dz/dt related if s is constant?

Textbook Question

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

b. How is ds/dt related to dy/dt and dz/dt if x is constant?