# CBSE Class 11 - Mathematics - Limits and Derivatives Part-7

**Derivatives Using First Principle**

**Theorem 1**: From the first principle, we have

Showing posts with label **class11-maths**. Show all posts

Showing posts with label **class11-maths**. Show all posts

**Derivatives Using First Principle**

In the previous post[Part-6], we learned about the Derivative of a function at a point. Let us focus on some important derivatives using the First Principle.

In the previous blog post Limits and derivatives Part-5, we learned about the limits. Let us understand the derivative of a function at a point.

**Q1: What is the derivative of a function at a point?**

Answer: Let y = f(x) is a continuous function. It means the value of y changes as the value of x changes.

At x = a, is a point in its domain of definittion. The derivative of f at a is defines as:

Let us revisit the basic terminology used in probability.

**Q1: Define probability. What can the probability never predict?**

Answer: Probability gives us a measure of the likelihood that something will happen. However, probability can never predict the number of times that an occurrence actually happens.

**Q2: Define experiment.**

Answer: An action or operation resulting in two or more well-defined outcomes.

e.g. tossing a coin, throwing a die, drawing a card from a pack of well-shuffled playing cards etc.

India celebrates National Mathematics Day on **December 22** every year. The day marks the birth anniversary of famous mathematician Srinivasa Ramanujan.

*Typo in the graphic new = knew

In the previous blog post Limits and derivatives Part-4, we learned about the trigonometric limits. Let us solve other problems related to Limits

In the previous blog post Limits and derivatives Part-3, we learn about the algebra of limits. Now, let us study few theorems before we discuss about trignometric limits.

For some a, if $\lim_{x\rightarrow a} f(x)$ and $\lim_{x\rightarrow a} g(x)$ exist then $\lim_{x\rightarrow a} f(x) \leq \lim_{x\rightarrow a} g(x)$

Let f(x) and g(x) be two real valued functions with same domain such that f(x) ≤ g(x) ≤ h(x) for all x in common domain.

For some a, if $\lim_{x\rightarrow a} f(x) = l = \lim_{x\rightarrow a} h(x)$, then $\lim_{x\rightarrow a} g(x) = l$

**Part-3 Algebra of Limits **

In the previous post Limits and derivatives Part-2, we get basic ideas of the algebra of limits and also learned about rules and properties of limits.

Let us try to solve few problems:

**Q1**: Evaluate the given limit $\lim_{r\rightarrow 1} \pi r^2$

**Answer**: $\lim_{r\rightarrow 1} \pi r^2 = \pi(1)^2 = \pi$

**Questions and Answers **#class11Maths #Limits

In the previous blog post Limits and derivatives Part-1 , we learn

$\lim_{x\rightarrow a} f(x) = l$ and it is called **limit of the function f(x)**

① The two ways x could approach a number an either from left or from right, i.e., all the values of x near a could be less than a or could be greater than a.

② In this case the right and left hand limits are different, and hence we say that the limit of f(x) as x tends to zero does not exist (even though the function is defined at 0).

**Part-1 - Questions and Answers**

**Q1: Define Calculus.**

Answer: Calculus is that branch of mathematics that mainly deals with the study of change in the value of a function as the points in the domain change.

👉Note: The chapter "Limits and Derivatives" is an introduction to Calculus.

👉Calculus is a Latin word meaning ‘pebble’. Ancient Romans used stones for counting.

**Q2: Who are called pioneers of Calculus (who invented Calculus)?**

Answer: Issac Newton (1642 - 1727) and G. W. Leibnitz(1646 - 1717).

Both of them were invented independently around the 17th century.

**Q3: What is the meaning of 'x tends to a' or x → a?**

Answer: When x tends to a (x → a), x is nearly close to a but never equals to a.

e.g. x → 3 means the value of x maybe 2.99 or 2.999 or 2.999...9 is very close to 3 but not exactly equal to 3. Similarly, x may be 3.01, 3.001, 3.0001... from the right side and gets closer to 3.

**No. of Questions**: 15

**Time**: 30 minutes

**Chapters**: Sets, Relations and Functions, Complex Numbers, Sequence and Series, Straight Lines, Limits and Statistics

**Q1**: The set {1, 2, 3, ...} is _______ set. Fill in the blank.

(a) null

(b) finite

(c) infinite

(d) singleton

**Q1: What are imaginary numbers?**

Answer: If the square of a given number is negative then such a number is called an imaginary number.

**Q2: Name the mathematician who was the first to introduce the symbol i (iota) for square root of -1.**

Answer: Euler.

**Q3: Evaluate i⁹ + i¹⁹**

**RELATION **

Let A and B be two nonempty sets. Then, a relation R from A to B is a subset of (A × B).

Thus, R is a relation from A to B ⇔ R ⊆ (A × B).

If (a, b) ∈ R then we say that ‘a is related to b‘ and we write, a R b.

If (a, b) ∉ R then ‘a is not related to b‘ and we write, $a \not \mathrel{R} b$.

**Q1: Let A = {-1, 2, 4} and B = {1, 3}. Show A × B as arrow diagram.**

**Chapter: Relations and Functions**

① Let A and B be two nonempty sets. Then, the Cartesian product of A and B is the set denoted by (A×B), consisting of all ordered pairs (a, b) such that a ∈ A and b ∈ B.

∴ A × B = {(a, b): a ∈ A and b ∈ B }.

② If A = ϕ or B = ϕ (empty sets), we define A × B = ϕ

③ B × A = {(b, a) : b ∈ B and a ∈ A} and A × A = {(a,b):a,b ∈ A}.

**Q1: What is an ordered pair?**

Answer: An ordered pair is a pair of entries in the specified order.

**Q2: What is ordered 2-tuple?**

Answer: Another name of an ordered pair.

**Q3: If A and B are any two sets, write to represent an ordered pair of elements of A and B? **

Answer: (a, b) : a ∈ A, b ∈ B

**Q4: Is (a, b) = (b, a)?**

Answer: No. a, b) ≠ (b, a) unless a = b

Learning is a continual process. To improve scores, the best way is to learn and practice. Solving question papers help students to evaluate their knowledge and familiarise with different types of question patterns being asked in the examination.

Eduvictors provides CBSE Previous Year Papers, Study Notes and Sample Question Papers for Class 11 Maths, Physics, Accounts, English, Hindi, Business Studies and Physical Education with/without solutions to help students in their board exam preparation.

1. Buy Oswaal CBSE Sample Question Paper Class 11 Mathematics Book (Reduced Syllabus for 2021 Exam)

Here attached the Maths Preboard Sample Question Paper (2020-2021) for your practice.

Answer: Given,

n( X ) = 17,

n( Y ) = 23 and

n( X ∪ Y ) = 38

We know that, n(X ∪ Y) = n(X) + n(Y) - n (X ∩ Y)

∴ 38 = 17 + 23 - n (X ∩ Y)

⇒ n (X ∩ Y) = 17 + 23 - 38 = 40 - 38 = 2

∴ n (X ∩ Y) = 2.

CBSE Class 11 Maths

(i) A'

(ii) B'

(iii) (A ∪ C)'

(iv) (A ∪ B)'

(v) (A')'

(vi) (B – C)'

(i) A' = U - A = {5,6,7,8,9}

(ii) B' = U - B = {1,3,5,7,9}

(i) A – B = {3, 6, 9, 15, 18, 21}

(ii) A – C = {3, 9, 15, 18, 21}

(iii) A – D = {3, 6, 9, 12, 18, 21}

(iv) B – A = {4, 8, 16, 20}

(v) C – A = {2,4,8,10,14,16}

(vi) D – A = {5,10,20}

(vii) B – C = {20}

(viii) B – D = {4,8,12,16}

(ix) C – B = {2,6,10,14}

(x) D – B = {5,10,15}

(xi) C – D = {2,4,6,8,12,14,16}

(xii) D – C = {5,15,20}

Here follows few important Laws of Set Operations

1. Idempotent Laws:

(a) A ∪ A = A

(b) A ∩ A = A

NCERT Solutions

Answer:

(i) A ∩ B = {7,9,11}

(ii) B ∩ C = {11, 13}

(iii) A ∩ C ∩ D = {A ∩ C} ∩ D = {11} ∩ {15, 17} = φ

(iv) A ∩ C = {11}

(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17} = φ

(vi) A ∩ (B ∪ C)

= {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

(vii) A ∩ D = φ

(viii) A ∩ (B ∪ D)

= {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {15, 17})

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17}

= {7, 9, 11}

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