Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.8.42a

Chapter 3, Problem 3.8.42a

Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
a. Find dr/dh for a cone with a lateral surface area of A=1500π.

Verified step by step guidance

Start with the formula for the lateral surface area of a cone: A = πr√(r² + h²). We are given that A = 1500π.

Set up the equation 1500π = πr√(r² + h²). Divide both sides by π to simplify: 1500 = r√(r² + h²).

To find dr/dh, we need to differentiate both sides of the equation with respect to h. First, express r in terms of h if possible, or use implicit differentiation.

Differentiate the equation 1500 = r√(r² + h²) with respect to h. Use the product rule and chain rule as necessary. The derivative of the left side is 0 since it's a constant.

Solve the resulting equation for dr/dh. This will involve isolating dr/dh on one side of the equation, which may require algebraic manipulation and simplification.

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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 cone is the area of the cone's curved surface, excluding the base. It is calculated using the formula A = πr√(r² + h²), where r is the radius and h is the height of the cone. Understanding this formula is essential for solving problems related to the surface area of cones.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated. In this context, it allows us to find the rate of change of one variable with respect to another, such as dr/dh, without explicitly solving for one variable in terms of the other. This is particularly useful when dealing with relationships defined by equations like the surface area of a cone.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a variable y depends on u, which in turn depends on x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is crucial for finding derivatives like dr/dh when multiple variables are involved.

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Intro to the Chain Rule

Related Practice

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

Textbook Question

45–50. Tangent lines Carry out the following steps. <IMAGE>

a. Verify that the given point lies on the curve.

x³+y³=2xy; (1, 1)

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

4x³ =y²(4−x); x=2 (cissoid of Diocles)

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Textbook Question

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

tan xy = x+y; (0,0)

Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 3t⁴; a= -2, 2

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