Question 1. Write the negation of each of the following statements:
(i) For every x ∈ N, x + 3 > 10
(ii) There exists x ∈ N, x + 3 = 10
Solution:
(i) The negation of the statement is “There exist x ∈ N, such that x + 3 <= 10.”
(ii) The negation of the statement is “There exist x ∈ N, such that x + 3 ≠ 10.”
Question 2. Negate each of the following statements:
(i) All the students complete their homework.
(ii) There exists a number which is equal to its square.
Solution:
(i) Some of the students did not complete their homework.
(ii) For all real numbers x, x2≠x.”
Summary
Exercise 31.4 covers the differentiation of various functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions. Students learn to apply differentiation rules and formulas to find derivatives. Understanding derivatives is crucial for calculus and its applications. Differentiation helps analyze functions and model real-world phenomena. Practice questions reinforce learning and application. Derivatives measure rates of change, essential for optimization and physics.
