Discrete Mathematics Answers

2.1 Differentiate f(x) = (2x3 + 3x2

)(x2 + 5x3 +5) using product rule. (5)

 2.2 Differentiate f(x) = (8x

3 + 4x )(2x

2 +5) using product rule. (5)

 2.3 Differentiate f(x) = 

5𝑥³+7𝑥

2𝑥²−4𝑥+5

using quotient rule. (5)

 2.4 Differentiate f(x) = 

𝑥³+4

4𝑥− 5

using quotient rule. (5)

 2.5 Differentiate (x5 + 6)5 using chain rule (5)

Prove that the following argument is valid. Write all the necessary steps

a)Rashid didn’t perform well in the subject, but he was present in every class.

b)Every student who completed all the assignments, performed well in the subject.

c)If a student did well in the subject, then they completed all the assignments

Lead to the conclusion:

Not everyone completed all the assignments

Prove that (a ∧ (b → ¬a)) → ¬b is a tautology.

Prove the following sentences using any prove method [ the most appropriate one]:

a) If r is rational and s is irrational, prove that 2r+s is irrational.

b) If t and s are integers and t × s is even, then t is even or s is even.

 Model two contextualized problems using binary trees both quantitatively and qualitatively

Let p, q and r be statements. Suppose you know that the statement form ((q → p) ∨ r) ∨ (∼ r ∧ p) is false. What can you conclude about the truth values of the three statement variables?

Let p, q and r be statements. Use the Laws of Logical Equivalence and the equivalence of → to a disjunction to show that: ∼ ((p ∨ (q →∼ r)) ∧ (r → (p∨ ∼ q))) ≡ (∼ p ∧ q) ∧ r.