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

Related Rates

Calculus

4. Applications of Derivatives

Related Rates

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2:29 minutes

Problem 3.8.19a

Textbook Question

Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.

a. How is dA/dt related to dθ/dt if a and b are constant?

Verified step by step guidance

Start by understanding the formula for the area of a triangle: A = (1/2)ab sin(θ). Here, a and b are constants, and θ is the variable that changes over time.

To find how dA/dt is related to dθ/dt, apply the chain rule of differentiation. Since a and b are constants, differentiate A with respect to θ: dA/dθ = (1/2)ab cos(θ).

Now, use the chain rule to relate dA/dt to dθ/dt. The chain rule states that dA/dt = (dA/dθ) * (dθ/dt).

Substitute the expression for dA/dθ from step 2 into the chain rule formula: dA/dt = (1/2)ab cos(θ) * dθ/dt.

This equation shows that the rate of change of the area with respect to time, dA/dt, is directly proportional to the rate of change of the angle with respect to time, dθ/dt, and depends on the cosine of the angle θ.

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Key Concepts

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

Derivative

The derivative represents the rate of change of a function with respect to a variable. In this context, dA/dt is the derivative of the area A with respect to time t, indicating how the area changes as the angle θ changes over time.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It allows us to find the derivative of a function with respect to an intermediate variable, such as θ, when the function is expressed in terms of another variable, like t. Here, it helps relate dA/dt to dθ/dt.

Trigonometric Functions

Trigonometric functions, such as sine, are essential in relating angles to side lengths in triangles. The function sinθ is used in the formula for the area of a triangle, and understanding its derivative is crucial for determining how changes in θ affect the area A when differentiating A = (1/2) ab sinθ.

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a. surface area change?

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).

a. Assuming that x, y, and z are differentiable functions of t, how is ds/dt related to dx/dt, dy/dt, and dz/dt?

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?

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).

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Textbook Question

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A = (1/2) ab sinθ.

b. How is dA/dt related to dθ/dt and da/dt if only b is constant?

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Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ isA = (1/2) ab sinθ.a. How is dA/dt related to dθ/dt if a and b are constant?

Key Concepts

Derivative

Chain Rule

Trigonometric Functions

Watch next

Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.

a. How is dA/dt related to dθ/dt if a and b are constant?