Solve the following differential equations using the substitutions given. (a) (b)

A rectangular tank with its base horizontal is filled with water to a depth h at time t = 0. Water leaks out of the tank from a small hole in the base at a rate proportional to the square root of the depth of the water. If the depth of water is ½h at time T, find […]

A researcher is investigating the spread of a certain disease in a town with a population of 3000 people. The researcher suggests that I, the number of people infected by the disease at time t days satisfies the differential equation , where k is a positive constant. (i) Given that I = 30 when t = 0, […]

(a)Without using a calculator, solve the inequality (b) Deduce the range of values of x that satisfies. Solutions: (a): x≤-6 or 1≤x<3 or x>3 (b): 0<x≤e-6 or e≤x<e3 or x>e3 (a) (b) lnx ≤ -6 x ≤ e-6 1≤ lnx <3 e≤x<e3 lnx>3 x>e3 […]

The use of the vertical line test determines if a line equation is a function. If the vertical line intersects only 1 point, the line equation is a function. If the vertical line intersects more than 1 point, the line equation is not a function.

The figure shows the curve x = y2-9. Find the area of the region bounded by the curve, y-axis and the line y=4

Find the total area enclosed by the curve y = x(x+3)(2-x), the x-axis and the line x=4 solution: 21(1/12) units x(2x – x2 + 6 – 3x) =6x – x3 – x2

Differentiate y=5x(e3x) with respect to x and hence find ∫x(e3x) dx Solution: 1/3(x)(e3x) – 1/9(e3x) + C Y = 5x e3x dy/dx = 5x(e3x)(3) + 5 e3x dy/dx = 15x(e3x) + 5(e3x) ∫15x(e3x) + 5(e3x) dx = 5x(e3x) ∫15x(e3x) dx + (5(e3x))/3 = 5x(e3x) 15∫x(e3x) dx = 5x(e3x) – 5/3(e3x) ∫x(e3x) dx = 1/3(x)(e3x) – 1/9(e3x) + […]