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NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
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Page 1 Miscellaneous Exercise Page 2 Miscellaneous Exercise Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 1 0 lim 1 1 1 1 1 Let . Accordingly, f x+h By first principle, ' lim h h ii f x x x x x h f x h f x fx h ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? 0 0 0 1 1 1 lim 1 1 1 lim 1 lim h h h h x h x h x h x x x h h x x h ? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? 0 0 0 2 1 lim 1 lim 1 lim 11 . h h h x x h h x x h h h x x h x x h xx x ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 (iii) Let f(x) = sin (x + 1). Accordingly, f x+h sin 1 By first principle, ' lim 1 lim sin 1 sin 1 1 1 1 1 1 lim 2cos sin 22 h h h xh f x h f x fx h x h x h x h x x h x h ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 1 2 2 lim 2cos sin 22 h x h h h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 sin 22 2 lim cos . 2 2 h h xh h ? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? Page 3 Miscellaneous Exercise Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 1 0 lim 1 1 1 1 1 Let . Accordingly, f x+h By first principle, ' lim h h ii f x x x x x h f x h f x fx h ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? 0 0 0 1 1 1 lim 1 1 1 lim 1 lim h h h h x h x h x h x x x h h x x h ? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? 0 0 0 2 1 lim 1 lim 1 lim 11 . h h h x x h h x x h h h x x h x x h xx x ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 (iii) Let f(x) = sin (x + 1). Accordingly, f x+h sin 1 By first principle, ' lim 1 lim sin 1 sin 1 1 1 1 1 1 lim 2cos sin 22 h h h xh f x h f x fx h x h x h x h x x h x h ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 1 2 2 lim 2cos sin 22 h x h h h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 sin 22 2 lim cos . 2 2 h h xh h ? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 0 0 2 sin 1 2 2 1 2 lim cos .lim [ 0 0] 22 2 h h h x h h As h h hh ? ? ?? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 2 0 2 sin cos .1 lim 1 2 cos 1 (iv) Let f x =cos x- . Accordingly, f x+h cos 88 By first principle, ' lim h h xx x x xh f x h f x fx h ?? ? ? ?? ? ? ? ? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? 0 0 1 lim cos cos 88 1 88 88 lim 2sin sin 22 h h x h x h x h x x h x h ?? ?? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 0 0 2 1 4 lim 2sin sin 22 sin 2 2 4 lim sin 2 2 h h xh h h h xh h ? ? ? ? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 00 sin 2 h 2 4 lim sin .lim As h 0 0 22 2 20 4 sin .1 2 hh h xh h x ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? sin 8 x ? ?? ? ? ? ?? ?? Page 4 Miscellaneous Exercise Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 1 0 lim 1 1 1 1 1 Let . Accordingly, f x+h By first principle, ' lim h h ii f x x x x x h f x h f x fx h ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? 0 0 0 1 1 1 lim 1 1 1 lim 1 lim h h h h x h x h x h x x x h h x x h ? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? 0 0 0 2 1 lim 1 lim 1 lim 11 . h h h x x h h x x h h h x x h x x h xx x ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 (iii) Let f(x) = sin (x + 1). Accordingly, f x+h sin 1 By first principle, ' lim 1 lim sin 1 sin 1 1 1 1 1 1 lim 2cos sin 22 h h h xh f x h f x fx h x h x h x h x x h x h ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 1 2 2 lim 2cos sin 22 h x h h h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 sin 22 2 lim cos . 2 2 h h xh h ? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 0 0 2 sin 1 2 2 1 2 lim cos .lim [ 0 0] 22 2 h h h x h h As h h hh ? ? ?? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 2 0 2 sin cos .1 lim 1 2 cos 1 (iv) Let f x =cos x- . Accordingly, f x+h cos 88 By first principle, ' lim h h xx x x xh f x h f x fx h ?? ? ? ?? ? ? ? ? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? 0 0 1 lim cos cos 88 1 88 88 lim 2sin sin 22 h h x h x h x h x x h x h ?? ?? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 0 0 2 1 4 lim 2sin sin 22 sin 2 2 4 lim sin 2 2 h h xh h h h xh h ? ? ? ? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 00 sin 2 h 2 4 lim sin .lim As h 0 0 22 2 20 4 sin .1 2 hh h xh h x ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? sin 8 x ? ?? ? ? ? ?? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 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) Solution 2: Let f(x) = x + a. Accordingly, f(x + h) = x + h + a By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 ' lim lim lim lim 1 1 h h h h f x h f x fx h x h a x a h h h ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ?? ? ? 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): ? ? r px q s x ?? ?? ?? ?? Solution 3: Let f(x) = ? ? r px q s x ?? ?? ?? ?? By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 2 ' rr f x px q s s px q xx r px q rx s s p x r px q rx s p x rr px q s p xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Page 5 Miscellaneous Exercise Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 1 0 lim 1 1 1 1 1 Let . Accordingly, f x+h By first principle, ' lim h h ii f x x x x x h f x h f x fx h ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? 0 0 0 1 1 1 lim 1 1 1 lim 1 lim h h h h x h x h x h x x x h h x x h ? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? 0 0 0 2 1 lim 1 lim 1 lim 11 . h h h x x h h x x h h h x x h x x h xx x ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 (iii) Let f(x) = sin (x + 1). Accordingly, f x+h sin 1 By first principle, ' lim 1 lim sin 1 sin 1 1 1 1 1 1 lim 2cos sin 22 h h h xh f x h f x fx h x h x h x h x x h x h ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 1 2 2 lim 2cos sin 22 h x h h h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? 0 sin 22 2 lim cos . 2 2 h h xh h ? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 0 0 2 sin 1 2 2 1 2 lim cos .lim [ 0 0] 22 2 h h h x h h As h h hh ? ? ?? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 2 0 2 sin cos .1 lim 1 2 cos 1 (iv) Let f x =cos x- . Accordingly, f x+h cos 88 By first principle, ' lim h h xx x x xh f x h f x fx h ?? ? ? ?? ? ? ? ? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? 0 0 1 lim cos cos 88 1 88 88 lim 2sin sin 22 h h x h x h x h x x h x h ?? ?? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 0 0 2 1 4 lim 2sin sin 22 sin 2 2 4 lim sin 2 2 h h xh h h h xh h ? ? ? ? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? 00 sin 2 h 2 4 lim sin .lim As h 0 0 22 2 20 4 sin .1 2 hh h xh h x ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? sin 8 x ? ?? ? ? ? ?? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 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) Solution 2: Let f(x) = x + a. Accordingly, f(x + h) = x + h + a By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 ' lim lim lim lim 1 1 h h h h f x h f x fx h x h a x a h h h ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ?? ? ? 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): ? ? r px q s x ?? ?? ?? ?? Solution 3: Let f(x) = ? ? r px q s x ?? ?? ?? ?? By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 2 ' rr f x px q s s px q xx r px q rx s s p x r px q rx s p x rr px q s p xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 2 2 pr qr pr ps x x x qr ps x ? ? ? ? ? ?? 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 nonzero constants and m and n are integers): (ax + b) (cx + d) 2 Solution 4: Let ? ? ? ? ' f x ax b ?? ? ? 2 cx d ? By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 2 2 2 2 2 2 2 2 2 ' 2 2 22 2 dd f x ax b cx d cx d ax b dx dx dd ax b c x cdx d cx d ax b dx dx d d d d d ax b c x cdx d cx d ax b dx dx dx dx dx ax b c x cd cx d a c ax b cx d a cx d ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 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): ax b cx d ? ? Solution 5: Let f(x) = ax b cx d ? ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? 2 ' dd cx d ax b ax b cx d dx dx fx cx d ? ? ? ? ? ? ?Read More
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FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| 1. What are limits and derivatives? |  |
Ans. Limits and derivatives are fundamental concepts in calculus. A limit is a value that a function approaches as its input approaches a certain value or infinity. Derivatives, on the other hand, measure the rate at which a function changes with respect to its input.
| 2. How do you find the limit of a function? | |
Ans. To find the limit of a function, you can evaluate the function at values approaching the desired input value. If the values approach the same value as the input value gets closer and closer, that value is the limit. Alternatively, you can use algebraic techniques, such as factoring or rationalizing the expression, to simplify the function and determine the limit.
| 3. What is the significance of limits and derivatives in real-life applications? | |
Ans. Limits and derivatives have numerous real-life applications. In physics, derivatives are used to calculate velocity and acceleration. In economics, derivatives are used to analyze rates of change in variables like demand and supply. In engineering, derivatives are used to determine the slope of a curve or the rate of change in a system. Limits are used to analyze the behavior of functions and determine properties such as continuity or the existence of asymptotes.
| 4. Can you explain the concept of differentiability in relation to derivatives? | |
Ans. Differentiability is a property of functions that determine if they have a derivative at a specific point. A function is said to be differentiable at a point if the derivative exists at that point. This means that the function has a well-defined slope at that point. If a function is not differentiable at a point, it means that the slope of the function is undefined or discontinuous at that point.
| 5. How do you calculate derivatives using the limit definition? | |
Ans. The limit definition of a derivative involves taking the limit of the difference quotient as the change in the input approaches zero. The difference quotient is the change in the function divided by the change in the input. By evaluating this limit, you can find the derivative of a function at a specific point. However, in practice, more efficient techniques such as the power rule, product rule, or chain rule are often used to calculate derivatives.
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