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Page No 301:

Question 1:

Evaluate the Given limit:

ANSWER:

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Question 2:

Evaluate the Given limit:

ANSWER:

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Question 3:

Evaluate the Given limit:

ANSWER:

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Question 4:

Evaluate the Given limit:

ANSWER:

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Question 5:

Evaluate the Given limit:

ANSWER:

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Question 6:

Evaluate the Given limit:

ANSWER:

Put x + 1 = y so that y → 1 as x → 0.

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Question 7:

Evaluate the Given limit:

ANSWER:

At x = 2, the value of the given rational function takes the form.

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Question 8:

Evaluate the Given limit:

ANSWER:

At x = 2, the value of the given rational function takes the form.

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Question 9:

Evaluate the Given limit:

ANSWER:

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Question 10:

Evaluate the Given limit: 

ANSWER:

At z = 1, the value of the given function takes the form.

Put so that z →1 as x → 1.

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Question 11:

Evaluate the Given limit:

ANSWER:

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Question 12:

Evaluate the Given limit:

ANSWER:

At x = –2, the value of the given function takes the form.

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Question 13:

Evaluate the Given limit:

ANSWER:

At x = 0, the value of the given function takes the form.

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Question 14:

Evaluate the Given limit:

ANSWER:

At x = 0, the value of the given function takes the form.

Page No 302:

Question 15:

Evaluate the Given limit:

ANSWER:

It is seen that x → π ⇒ (π – x) → 0

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Question 16:

Evaluate the given limit: 

ANSWER:

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Question 17:

Evaluate the Given limit:

ANSWER:

At x = 0, the value of the given function takes the form.

Now,

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Question 18:

Evaluate the Given limit:

ANSWER:

At x = 0, the value of the given function takes the form.

Now,

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Question 19:

Evaluate the Given limit:

ANSWER:

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Question 20:

Evaluate the Given limit:

ANSWER:

At x = 0, the value of the given function takes the form.

Now,

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Question 21:

Evaluate the Given limit:

ANSWER:

At x = 0, the value of the given function takes the form.

Now,

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Question 22:

ANSWER:

At, the value of the given function takes the form.

Now, put  so that.

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Question 23:

Find f(x) andf(x), where f(x) =

ANSWER:

The given function is

f(x) =

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Question 24:

Find f(x), where f(x) =

ANSWER:

The given function is

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Question 25:

Evaluatef(x), where f(x) = 

ANSWER:

The given function is

f(x) = 

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Question 26:

Findf(x), where f(x) =

ANSWER:

The given function is

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Question 27:

Findf(x), where f(x) =

ANSWER:

The given function is f(x) =.

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Question 28:

Suppose f(x) = and iff(x) = f(1) what are possible values of a and b?

ANSWER:

The given function is

Thus, the respective possible values of a and b are 0 and 4.

Page No 303:

Question 29:

Letbe fixed real numbers and define a function

What isf(x)? For some  computef(x).

ANSWER:

The given function is.

Page No 303:

Question 30:

If f(x) =.

For what value (s) of a does f(x) exists?

ANSWER:

The given function is

When a < 0,

When a > 0

Thus,  exists for all a ≠ 0.

Page No 303:

Question 31:

If the function f(x) satisfies, evaluate.

ANSWER:

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Question 32:

If. For what integers m and n does  and exist?

ANSWER:

The given function is

Thus,  exists if m = n.

Thus, exists for any integral value of m and n.

Page No 312:

Question 1:

Find the derivative of x2 – 2 at x = 10.

ANSWER:

Let f(x) = x2 – 2. Accordingly,

Thus, the derivative of x2 – 2 at x = 10 is 20.

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Question 2:

Find the derivative of 99x at x = 100.

ANSWER:

Let f(x) = 99x. Accordingly,

Thus, the derivative of 99x at x = 100 is 99.

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Question 3:

Find the derivative of x at x = 1.

ANSWER:

Let f(x) = x. Accordingly,

Thus, the derivative of x at x = 1 is 1.

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Question 4:

Find the derivative of the following functions from first principle.

(i) x3 – 27 (ii) (x – 1) (x – 2)

(ii)  (iv) 

ANSWER:

(i) Let f(x) = x3 – 27. Accordingly, from the first principle,

(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,

(iii) Let. Accordingly, from the first principle,

(iv) Let. Accordingly, from the first principle,

Page No 312:

Question 5:

For the function

Prove that 

ANSWER:

The given function is

Thus,

Page No 313:

Question 6:

Find the derivative offor some fixed real number a.

ANSWER:

Let 

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Question 7:

For some constants a and b, find the derivative of

(i) (x – a) (x – b) (ii) (ax2 + b)2 (iii) 

ANSWER:

(i) Let f (x) = (x – a) (x – b)

(ii) Let 

(iii) 

By quotient rule,

Page No 313:

Question 8:

Find the derivative offor some constant a.

ANSWER:

By quotient rule,

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Question 9:

Find the derivative of

(i)  (ii) (5x3 + 3x – 1) (x – 1)

(iii) x–3 (5 + 3x) (iv) x5 (3 – 6x–9)

(v) x–4 (3 – 4x–5) (vi) 

ANSWER:

(i) Let

(ii) Let f (x) = (5x3 + 3x – 1) (x – 1)

By Leibnitz product rule,

(iii) Let f (x) = x– 3 (5 + 3x)

By Leibnitz product rule,

(iv) Let f (x) = x5 (3 – 6x–9)

By Leibnitz product rule,

(v) Let f (x) = x–4 (3 – 4x–5)

By Leibnitz product rule,

(vi) Let f (x) = 

By quotient rule,

Page No 313:

Question 10:

Find the derivative of cos x from first principle.

ANSWER:

Let f (x) = cos x. Accordingly, from the first principle,

Page No 313:

Question 11:

Find the derivative of the following functions:

(i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x

(iv) cosec x (v) 3cot x + 5cosec x

(vi) 5sin x – 6cos x + 7 (vii) 2tan x – 7sec x

ANSWER:

(i) Let f (x) = sin x cos x. Accordingly, from the first principle,

(ii) Let f (x) = sec x. Accordingly, from the first principle,

(iii) Let f (x) = 5 sec x + 4 cos x. Accordingly, from the first principle,

(iv) Let f (x) = cosec x. Accordingly, from the first principle,

(v) Let f (x) = 3cot x + 5cosec x. Accordingly, from the first principle,

From (1), (2), and (3), we obtain

(vi) Let f (x) = 5sin x – 6cos x + 7. Accordingly, from the first principle,

(vii) Let f (x) = 2 tan x – 7 sec x. Accordingly, from the first principle,

Page No 317:

Question 1:

Find the derivative of the following functions from first principle:

(i) –x (ii) (–x)–1 (iii) sin (x + 1)

(iv) 

ANSWER:

(i) Let f(x) = –x. Accordingly,

By first principle,

(ii) Let. Accordingly,

By first principle,

(iii) Let f(x) = sin (x + 1). Accordingly,

By first principle,

(iv) Let. Accordingly,

By first principle,

Page No 317:

Question 2:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)

ANSWER:

Let f(x) = x + a. Accordingly,

By first principle,

Page No 317:

Question 3:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

By Leibnitz product rule,

Page No 317:

Question 4:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b) (cx + d)2

ANSWER:

Let

By Leibnitz product rule,

Page No 317:

Question 5:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

Page No 317:

Question 6:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

ANSWER:

By quotient rule,

Page No 317:

Question 7:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

Page No 317:

Question 8:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

By quotient rule,

Page No 317:

Question 9:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

By quotient rule,

Page No 317:

Question 10:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Page No 317:

Question 11:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Page No 317:

Question 12:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n

ANSWER:

By first principle,

Page No 317:

Question 13:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n (cx + d)m

ANSWER:

Let

By Leibnitz product rule,

Therefore, from (1), (2), and (3), we obtain

Page No 317:

Question 14:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin (x + a)

ANSWER:

Let

By first principle,

Page No 317:

Question 15:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

ANSWER:

Let

By Leibnitz product rule,

By first principle,

Now, let f2(x) = cosec x. Accordingly,

By first principle,

From (1), (2), and (3), we obtain

Page No 317:

Question 16:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

Page No 318:

Question 17:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

Page No 318:

Question 18:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

Page No 318:

Question 19:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinn x

ANSWER:

Let y = sinn x.

Accordingly, for n = 1, y = sin x.

For n = 2, y = sin2 x.

For n = 3, y = sin3 x.

We assert that 

Let our assertion be true for n = k.

i.e., 

Thus, our assertion is true for n = k + 1.

Hence, by mathematical induction,

Page No 318:

Question 20:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

By quotient rule,

Page No 318:

Question 21:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

By first principle,

From (i) and (ii), we obtain

Page No 318:

Question 22:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x4 (5 sin x – 3 cos x)

ANSWER:

Let

By product rule,

Page No 318:

Question 23:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x2 + 1) cos x

ANSWER:

Let

By product rule,

Page No 318:

Question 24:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax2 + sin x) (p + q cos x)

ANSWER:

Let

By product rule,

Page No 318:

Question 25:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By product rule,

Let. Accordingly,

By first principle,

Therefore, from (i) and (ii), we obtain

Page No 318:

Question 26:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

Page No 318:

Question 27:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

ANSWER:

Let

By quotient rule,

Page No 318:

Question 28:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By first principle,

From (i) and (ii), we obtain

Page No 318:

Question 29:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x – tan x)

ANSWER:

Let

By product rule,

From (i), (ii), and (iii), we obtain

Page No 318:

Question 30:

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 

ANSWER:

Let

By quotient rule,

It can be easily shown that 

Therefore,