Description
2] 1. (a) Sketch the graph of f(x) = 4 − x2 on the interval [−1, 2], and approximate the area between the graph and the x-axis on [−1, 2] by the left Riemann sum L3 using partitioning of the interval into 3 subintervals of equal length.
(b) For the same f(x) = 4 − x, write in sigma notation the formula for the left Riemann sum Ln with partitioning of the interval [−1, 2] into n subintervals of equal length, and calculate R 2 −1 f(x) dx as the limit of Ln at n → ∞ NOTE: you may need the formulas Pn
k=1
k =
n(n+1)
2
,Pn
k=1
k
2 =
n(n+1)(2n+1)
6
.
(c) Calculate the derivative of the function F(x) = sec(3x) + tan(3x)R0e−t2
dt(Hint: use the Fundamental Theorem of Calculus and differentiation rules.)
[12] 2. Evaluate the following definite integrals (give the exact answers):
(a)Z30x√9 − x2 dx (b)Ze1ln2x dx
[6] 3. Find F(t) such that F
0(t) = sec4(t) and Fπ4= 0.
[10] 4. Calculate the following indefinite integrals:
(a)Z(x2 − 2x) sin(2x) dx (b)Zx2 + 3×2 − 3xdx
[8] 5. Evaluate the given improper integral or show that it diverges:
(a)Z∞0x2e−x3dx (b)Z10xx2 − 1dx
[17] 6. (a) Sketch the curves y = √2x and y = x and find the area enclosed.
(b) Sketch the region enclosed by the parabola x = y2 + 1 and the line x = 5 and find the volume of the solid obtained by revolution of this region about the line x = 5.
(c) Find the average value of the function f(x) = x√1 + 2x on the interval [0, 4].
[9] 7. Find the limit of the sequence {an} or prove that the limit does not exist:
(a) an =3n − n22n
(b) an =ln(n3)n + 1
(c) an =√n + 100 −√n
[8] 8. Determine whether the series is divergent or convergent, and if convergent,
then absolutely or conditionally :
(a)X∞n=2(−1)nln nn(b)X∞n=0(−1)n+1 n + 100100n + 1
[10] 9. Find the radius and the interval of convergence of the following series
(a)X∞1(3x)nn!
(b)X∞n=1(x + 1)3nn 8n
[8] 10. (a) Derive the Maclaurin series of f(x) = x3ln(1 + 2×2)
(HINT: start with the series for ln(1 + z) where z = 2×2).
(b) Use differentiability of power series to find the sum
F(x) = P∞1(x − 1)nn
within its radius of convergence.
[5] Bonus Question. A solid is generated by rotating about the x-axis the region enclosed
between the curve y = f(x) and x-axis on the interval [0, b], where f is a positive function and
x ≥ 0. For all values of b ≥ 0 the generated solid has the volume πb4 . Find the function f.
