Find c , so that f ' ( c ) = f ( b ) - f ( a ) b - a in the following case f ( x ) = x 2 - 3 x - 1 ; a = - 11 7 , b = 13 7

Given, f ( x ) = x 2 - 3 x - 1 ; a = - 11 7 , b = 13 7 Now, f ( b ) = f 13 7 = 13 7 2 - 3 13 7 - 1 = 169 49 - 3 ( 13 ) 7 - 1 = 169 - 273 - 49 49 = - 153 49 f ( a ) = f - 11 7 = - 11 7 2 - 3 - 11 7 - 1 = 121 49 - 3 ( - 11 ) 7 - 1 = 121 + 231 - 49 49 = 303 49 Now, f ' ( x ) = 2 x - 3 ⇒ f ' ( c ) = 2 c - 3 Given f ' ( c ) = f ( b ) - f ( a ) b - a 2 c - 3 = - 153 49 - 303 49 13 7 + 11 7 = - 456 49 24 7 2 c = - 19 7 + 3 = 2 7 ⇒ c = 1 7 Therefore, value of c = 1 7 .

Find c , so that f ' ( c ) = f ( b ) - f ( a ) b - a in the following case f ( x ) = e x ; a = 0 , b = 1

Given that, f x = e x , a = 0 , b = 1 Now, f ( b ) = f 1 = e 1 = e f ( a ) = f 0 = e 0 = 1 f ' ( x ) = e x f ' ( c ) = e c Given f ' ( c ) = f ( b ) - f ( a ) b - a Substitute the values in f ' c ⇒ e c = e - 1 1 Apply log on both sides log e c = log e - 1 ∴ c = log e - 1 Therefore, the value of the c = log e - 1

E M B I B E

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(h)>Q III

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Earn 100

If the curved surface of right circular cylinder inscribed in a sphere of radius $R$ is maximum, show that the height of the cylinder is $\sqrt{2} R$ .

Important Questions on Applications of Derivatives

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(a)>Q II

Find the approximations of the following

\sqrt{82}

.(Answer up to

3

decimals)

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(a)>Q II

Find the approximations of the following

\cos (60^{0} 5^{'})

.(Answer up to

\Rightarrow h^{2} = 2 R^{2} \Rightarrow h = \sqrt{2} R

decimals)

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(c)>Q II

Find the value of

k

so that the length of the sub-normal at any point on the curve

x \cdot y^{k} = a^{k + 1}

is constant.

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(e)>Q I

If

y = x^{4}

then find the average rate of change of

y

between

x = 2

and

x = 4

.

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(f)>Q I

Find

c

, so that

f^{'} (c) = \frac{f (b) - f (a)}{b - a}

in the following case

f (x) = x^{2} - 3 x - 1; a = \frac{- 11}{7}, b = \frac{13}{7}

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(f)>Q I

Find

c

, so that

f^{'} (c) = \frac{f (b) - f (a)}{b - a}

in the following case

f (x) = e^{x}; a = 0, b = 1

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(h)>Q I

Find the points of local extrema (if any) and local extrema of the following function each of whose domain is shown against the function

f (x) = (x - 1) (x + 2)^{2} \forall x \in ℝ

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 10 - Applications of Derivatives>Exercise 10(h)>Q III

A wire of length

l

is cut into two parts which are bead respectively in the form of a square and a circle. What are the lengths of the pieces of the wire respectively so that the sum of the areas is the least.