yA r = f(8) P(r, y) = P(r, 0)

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Tangents and normals Let a polar curve be described by r = ƒ(θ), and let be the line tangent to the curve at the point P(x, y) = P(r, θ) (see figure).
a. Explain why tan a = dy/dx.
b. Explain why tan θ = y/x.
c. Let φ be the angle between and the line through O and P.
Prove that tan φ = ƒ(θ)/ƒ'(θ).
d. Prove that the values of u for which is parallel to the x-axis
satisfy tan θ = -ƒ(θ)/ƒ'(θ).
e. Prove that the values of u for which is parallel to the y-axis
satisfy tan θ = ƒ(θ)/ƒ'(θ).

yA
r = f(8)
P(r, y) = P(r, 0)
Transcribed Image Text:yA r = f(8) P(r, y) = P(r, 0)
Expert Solution
Step 1

Consider the given figure:

Calculus homework question answer, step 1, image 1

The line l is tangent to the curve. The tangent line is making alpha angle with the positive x axis. This gives the slope of the line.

The slope of the line is the rate of change of dependent variable with respect to the independent variable.

Therefore, It is said that 

Calculus homework question answer, step 1, image 2

 

Step 2

Consider the given figure:

Calculus homework question answer, step 2, image 1

The triangle formed by the point P with the coordinate axis is a right angled triangle.

The tangent of an angle in a triangle is equal to the ratio of perpendicular to the base.

Thus, it is said that 

Calculus homework question answer, step 2, image 2

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