Textbook Question
Consider the function
f(x) = { x² cos(2/x), x ≠ 0
0, x = 0
b. Determine f' for x ≠ 0.
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.9.48b
Tolerance
b. About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?
Verified step by step guidance1
Identify the formula for the surface area of the side of a cylindrical tank, which is given by \( A = 2\pi rh \), where \( r \) is the radius and \( h \) is the height of the cylinder.
Recognize that the diameter \( D \) is related to the radius by \( D = 2r \). Therefore, the surface area can also be expressed in terms of the diameter as \( A = \pi Dh \).
To find the tolerance in the diameter measurement, we need to determine how a small change in \( D \) affects the surface area \( A \). This involves calculating the derivative \( \frac{dA}{dD} \), which represents the rate of change of the area with respect to the diameter.
Calculate the derivative \( \frac{dA}{dD} = \pi h \). This tells us how much the area changes for a small change in the diameter.
Set up the inequality \( \left| \frac{\Delta A}{A} \right| \leq 0.05 \) to ensure the change in area is within 5% of the true area. Substitute \( \Delta A = \pi h \Delta D \) and solve for \( \Delta D \) to find the required accuracy in the diameter measurement.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Approximation
Differential approximation is a method used in calculus to estimate how a small change in one variable affects a function's output. It involves using the derivative of the function to approximate the change in the function's value. This concept is crucial for understanding how small errors in measurement can impact the calculated amount of paint needed for the tank.
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05:53
Finding Differentials
Surface Area of a Cylinder
The surface area of a cylinder, particularly the lateral surface area, is calculated using the formula 2πrh, where r is the radius and h is the height. Understanding this formula is essential for determining how changes in the tank's diameter affect the surface area, and consequently, the amount of paint required.
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09:07Example 1: Minimizing Surface Area
Percentage Error
Percentage error quantifies the accuracy of a measurement by comparing the error to the true value, expressed as a percentage. In this context, it helps determine how precise the diameter measurement must be to ensure the paint calculation is within 5% of the true amount. This concept is vital for assessing the impact of measurement inaccuracies on the final result.
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04:57Determining Error and Relative Error
Related Practice
Textbook Question
Find y⁽⁴⁾ = d⁴y/dx⁴ if:
b. y = 9 cos x
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Textbook Question
Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation
______
S = πr √ r² + h².
b. How is dS/dt related to dh/dt if r is constant?
Textbook Question
Analyzing Motion Using Graphs
[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:
b. When does it move to the left (down) or to the right (up)?
s = 200t - 16t², 0 ≤ t ≤ 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft/sec)
Textbook Question
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.
b. How is dS/dt related to dh/dt if r is constant?
Textbook Question
b. Show that
f(x) = { x² sin(1/x), x ≠ 0
0, x = 0
is differentiable at x = 0 and find f′(0).