Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Implicit Differentiation

Calculus

4. Applications of Derivatives

Implicit Differentiation

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1:54 minutes

Problem 3.8.42b

Textbook Question

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.
b. Evaluate this derivative when r=30 and h=40.

Verified step by step guidance

First, identify the formula for the lateral surface area of a cone, which is given as A = πr√(r² + h²).

To find the derivative of A with respect to r, apply the chain rule. The expression inside the square root, r² + h², is a function of r, so differentiate it with respect to r.

The derivative of r² with respect to r is 2r, and since h is a constant, the derivative of h² with respect to r is 0. Therefore, the derivative of r² + h² with respect to r is 2r.

Now, differentiate the entire expression A = πr√(r² + h²) with respect to r. Use the product rule, which states that the derivative of a product u*v is u'v + uv'. Here, u = πr and v = √(r² + h²).

Evaluate the derivative at r = 30 and h = 40 by substituting these values into the derivative expression obtained in the previous step.

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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. This formula derives from the geometry of the cone and involves the Pythagorean theorem to account for the slant height.

Derivative

A derivative represents the rate of change of a function with respect to a variable. In this context, it helps determine how the lateral surface area A changes as the radius r varies, while keeping the height h constant. The derivative is calculated using differentiation rules, which provide insights into the behavior of the function at specific points.

Evaluation of Derivatives

Evaluating a derivative involves substituting specific values into the derivative function to find the instantaneous rate of change at those points. In this case, after finding the derivative of the lateral surface area with respect to r, we will substitute r = 30 and h = 40 to compute the exact rate of change of the surface area at that radius.