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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.8.19a
Chapter 3, Problem 3.8.19a

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?

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1
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.
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Derivatives

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.
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Intro to the Chain Rule

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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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.


a. How fast is the boat approaching the dock when 10 ft of rope are out?


Textbook Question

Differentiability and Continuity on an Interval


Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be


a. differentiable?


Give reasons for your answers.

Textbook Question

Computer Explorations


Use a CAS to perform the following steps in Exercises 55–62.


a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.


2y² + (xy)¹/³ = x² + 2, P(1,1)

Textbook Question

a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).

Textbook Question

The accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.


a. When does the body reverse direction?

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?

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