Textbook Question
Find y⁽⁴⁾ = d⁴y/dx⁴ if:
b. y = 9 cos x
2
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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 3.96b
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?
Verified step by step guidance1
Start with the given formula for the lateral surface area of a right circular cone: \( S = \pi r \sqrt{r^2 + h^2} \).
Since the radius \( r \) is constant, differentiate both sides of the equation with respect to time \( t \) to find \( \frac{dS}{dt} \).
Apply the chain rule to differentiate the right side: \( \frac{dS}{dt} = \pi r \cdot \frac{d}{dt}(\sqrt{r^2 + h^2}) \).
Differentiate \( \sqrt{r^2 + h^2} \) with respect to \( t \) using the chain rule: \( \frac{d}{dt}(\sqrt{r^2 + h^2}) = \frac{1}{2\sqrt{r^2 + h^2}} \cdot 2h \cdot \frac{dh}{dt} \).
Substitute the derivative back into the expression for \( \frac{dS}{dt} \): \( \frac{dS}{dt} = \pi r \cdot \frac{h}{\sqrt{r^2 + h^2}} \cdot \frac{dh}{dt} \). This shows how \( \frac{dS}{dt} \) is related to \( \frac{dh}{dt} \) when \( r \) is constant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Lateral Surface Area of a Cone
The lateral surface area of a right circular cone is the area of the cone's curved surface, excluding the base. It is calculated using the formula S = πr√(r² + h²), where r is the radius of the base and h is the height of the cone. Understanding this formula is essential for analyzing how changes in height or radius affect the surface area.
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Example 1: Minimizing Surface Area
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 rate of change of the lateral surface area (dS/dt) is related to the rate of change of height (dh/dt) while keeping the radius constant. This concept is fundamental in calculus for solving problems involving dynamic systems.
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04:16Intro To Related Rates
Differentiation
Differentiation is a key concept in calculus that involves finding the derivative of a function, which represents the rate of change of that function with respect to a variable. In this problem, we will differentiate the lateral surface area formula with respect to time to establish the relationship between dS/dt and dh/dt. Mastery of differentiation techniques is crucial for solving related rates problems.
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05:53Finding Differentials
Related Practice
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
The Reciprocal Rule
b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.
Textbook Question
Recovering a function from its derivative
b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).
Textbook Question
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?
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).