Solution Manual Exam Questions (40 Terms) With Verified Definitions Updated 2024

Find the area of the polar equation r = sin2θ + 2 {Ans: A = (1/2) ∫ (sin2θ + 2)^2 dθ from [0, 2pi] plug into calculator A = 9pi/4}Is the function continuous at the point x = 2? f(x) = x+1, x < 2 f(x) = 2x+1, x≥2 {Ans: Condition 1: Does f(2) exist? Yes 2(2) +1 = 5 Condition 2: Does lim f(x) as x approaches 2 exist? Left limit is 3, Right limit is 5 Because the two limits don't match the limit does not exist so the function is not continuous at x = 2.}∫(x-18)/(x+3)(x-4) dx = {Ans: Use partial fractions to evaluate the integral. First, we write the intragrand as: (x-18)/(x+3)(x-4)= A/(x+3) + B/(x-4) Multiply both sides by: (x+3)(x-4) Now we have: x-18 = A(x-4)+B(x+3) x-18= Ax-4A+Bx+3B x-18= (Ax+Bx) + (3B-4A) x-18= x(A+B) + (3B-4A) Thus (A+B)=1 and (3-4A)= -18 Solve this using simultaneous equations to get A=3 and B=-2. We can now rewrite the original integral as: ∫(3/(x+3)) + (-2/(x-4))dx = ∫3dx/(x+3) - ∫2dx/(x-4)}Use the trapezoid rule with n = 4 to approximate the area between the curve f(x) = x^3 -x and the x-axis from x = 3 to x