Page No 301:
Question 1:
Evaluate the Given limit:
Answer:
Question 2:
Evaluate the Given limit:
Answer:
Question 3:
Evaluate the Given limit:
Answer:
Question 4:
Evaluate the Given limit:
Answer:
Question 5:
Evaluate the Given limit:
Answer:
Question 6:
Evaluate the Given limit:
Answer:
Put x + 1 = y so that y → 1 as x → 0.
Question 7:
Evaluate the Given limit:
Answer:
At x = 2, the value of the given rational function takes the form.
Question 8:
Evaluate the Given limit:
Answer:
At x = 2, the value of the given rational function takes the form.
Question 9:
Evaluate the Given limit:
Answer:
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.
Question 11:
Evaluate the Given limit:
Answer:
Question 12:
Evaluate the Given limit:
Answer:
At x = –2, the value of the given function takes the form.
Question 13:
Evaluate the Given limit:
Answer:
At x = 0, the value of the given function takes the form.
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
Question 16:
Evaluate the given limit:
Answer:
Question 17:
Evaluate the Given limit:
Answer:
At x = 0, the value of the given function takes the form.
Now,
Question 18:
Evaluate the Given limit:
Answer:
At x = 0, the value of the given function takes the form.
Now,
Question 19:
Evaluate the Given limit:
Answer:
Question 20:
Evaluate the Given limit:
Answer:
At x = 0, the value of the given function takes the form.
Now,
Question 21:
Evaluate the Given limit:
Answer:
At x = 0, the value of the given function takes the form.
Now,
Question 22:
Answer:
At, the value of the given function takes the form.
Now, put so that.
Question 23:
Find f(x) andf(x), where f(x) =
Answer:
The given function is
f(x) =
Question 24:
Find f(x), where f(x) =
Answer:
The given function is
Question 25:
Evaluatef(x), where f(x) =
Answer:
The given function is
f(x) =
Question 26:
Findf(x), where f(x) =
Answer:
The given function is
Question 27:
Findf(x), where f(x) =
Answer:
The given function is f(x) =.
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.
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.
Question 31:
If the function f(x) satisfies, evaluate.
Answer:
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.
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.
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.
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,
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
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,
Question 8:
Find the derivative offor some constant a.
Answer:
By quotient rule,
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,
Question 10:
Find the derivative of cos x from first principle.
Answer:
Let f (x) = cos x. Accordingly, from the first principle,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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:
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:
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,
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
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,
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
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,
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,
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,
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,
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
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,
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,
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,
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
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,
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,
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
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
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,
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