Calculus Answers

1. find the domains of f(x) =

√x+7 3

______  and g(x) =   _______ 

 x+4   |x| + 2

1. find the domains of f(x)= √x+7 / x+4= g(x) = 3 / |x| + 2 
2. Draw the graph of f(x) =x³. then use it together with some transformation to sketch the graph of g(x) =(x - 3)³ - 4.

The population 𝑃 of a town decreases at a rate proportional to the number by which the population exceeds a 1000. (i) Form the differential equation involving 𝑃, (ii) Show that 𝑃 = 1000 + 𝐴𝑒 −𝑘𝑡, where 𝐴 and 𝑘 are constants and 𝑃 is the solution of the differential equation, (iii) Initially, the population of the town was 2500. Ten years later, it had fallen to 1,900. In how many years will the population be 1500? (iv) What does the mathematical model predict about the population of the town in the long term?

Solve the following differential equations: (i) (𝑦 3 + 1) 𝑑𝑦 𝑑𝑥 − 𝑥𝑦 2 = 𝑥, (ii) 𝑥𝑦 2 𝑑𝑦 𝑑𝑥 = 𝑥 3 − 𝑦 3, (iii) (1 + sin 𝑥) 𝑑𝑦 𝑑𝑥 + 𝑦 cos 𝑥 = tan 𝑥, (iv) 𝑥 𝑑𝑦 𝑑𝑥 − 5𝑦 = 𝑥 7.

Show that "\\int" 𝑥 3 cos 𝑥 𝑑𝑥 𝜋 2 0 = 𝜋 3 8 − 3𝜋 + 6

Find (i) ∫ ln 𝑥 𝑑𝑥, (ii) ∫ 𝑒 −𝑥 cos 2𝑥 𝑑𝑥. Hint: Let 𝐼 = ∫ 𝑒 𝑥 cos 2𝑥 𝑑𝑥 and use parts twice, (iii) sin−1 𝑥 𝑑𝑥.

Question 1 Find (i) ∫ 𝑑𝑥 𝑥(ln 𝑥)2, (ii) ∫ 𝑥√𝑥 + 1 𝑑𝑥, (iii) ∫ 𝑥(𝑥 2 + 3) 4𝑑𝑥 1 0 , (iv) cos2 𝑥 sin3 𝑥 𝑑𝑥. Try 𝑡 = sin x