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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Using a Right Triangle Consider the right triangle shown in the figure.

A ( α ) = 1 2 0 α sec 2 θ d θ .

α = 0 α sec 2 δ d θ .

(c) Use part (b) to derive the formula for the derivative of the tangent function.

Chapter 10, Problem 10PS, Using a Right Triangle Consider the right triangle shown in the figure. A()=120sec2d. =0sec2d. (c)

Figure for 10

(a)

To determine

To prove: The area of the given triangle, refer to the figure in the question, is A(α)=120αsec2θdθ.

Explanation

Given:

Refer to the figure provided in the question.

Formula used:

The area of the polar curve is given by, 120αr2dθ where r is the function of θ.

Proof:

Consider the figure provided in the question.

Now, the area of the triangle is given by, 120αr2dθ.

Now, in the triangle, assume the hypotenuse to be r

(b)

To determine

To prove: The statement, tanα=0αsec2θdθ with the help of part (a).

Now, first consider the figure given in the question.

Area of the triangle is equal to 12×base×height.

Now, refer to the provided figure in the question.

So,

tanα=heightbasetanα=height1height=tanα

So, area of the triangle would become, 12tanα.

Now, from part (a) the area of the triangle is, 120α(secα)2dθ.

So, put 12tanα equal to 120α(secα)2dθ.

12tanα=120α(secα)2dθtanα=0α(secα)2dθ

Hence proved.

(c)

To determine

The value of the differentiation of the tangent function from part (b).

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