Q1. Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
(i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}
(ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}
(iii) {(1,3), (1,5), (2,5)}.
Q1. Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.
RELATIONS AND FUNCTIONS
Q1. If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.
Answer: Since ordered pairs are equal,
∴ x/3 + 1 = 5/3
⇒ x/3 = 5/3 - 1 = 2/3
⇒ x = 2
y - 2/3 = 1/3
y = 1/3 + 2/3
y = 3/3 = 1
Thus, x = 2 and y = 1
You should remember
1. In the sexagesimal system, we measure angles in degrees, minutes and seconds.
1 right angle = 90°
1° = 60′ (minute) and
1′ = 60′′ (second)
2. In the circular measure, we measure angles in radians.
Thus,
πc = 180°
CBSE Class 11 - Mathematics - Limits and Derivatives Part-11
Working rule for finding the derivative of implicit functions
First method:
Differentiate every term of f (x, y) = 0 with respect to x.
CBSE Class 11 - Mathematics - Limits and Derivatives Part-10
You have learnt theorems of differentiation part-9. Let us proceed further. Here is a list of differentiation of some standard functions:
CBSE Class 11 - Mathematics - Limits and Derivatives Part-9
You have learnt about finding derivatives using the first principle in part-7 and part-8. Let us proceed further. Here is a summary of theorems on differentiation.
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| Differentiation |
Derivatives Using First Principle
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.
*Typo in the graphic new = knew
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.
