Calculus Answers

Four bugs are placed at the four corners of a square with side length a. The bugs crawl counter-clockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the centre of the square along spiral paths.
a) Find the polar equation of a bug's path assuming the origin (pole) is at the centre of the square. (Use the fact that the line joining one bug to the next is tangent to the bug's path).
b) Find the distance travelled by a bug by the time it meets the other bugs at the centre.

For each of the following functions, find the first three derivatives (without simplifying) then generalise the pattern to find a formula for the nth derivative of f. Evaluate the nth derivative for the value of x given.
a) f(x)=x^(2/3), x=1
b) f(x)=ln(1+2x), x=0

Determine the convergence or divergence of the sequence of partial sums whose nth term is given by:
n
a) un= ∑ 1/(2k+1) - 1/(2k+3)
k=1

b) un=n^2 e^-n

Find the values of a and b that make f continuous for x∈ℝ
f(x) = {(x^2 -4)/(x-2), if x<2;
ax^2 -bx+3, if 2≤x<3;
2x-a+b, if x≥3

Determine whether the sequence converges or diverges. If it converges, find the limit.
a) an = (n^3 -1)/(2n^3 +1)
b) {ln(n)/ln(2n)}
c) an = (1+ 2/n)^n

Consider the slope and point given below.
m = 1/3 , P(−3, 8)

Find the slope-intercept form of the line with the given slope which passes through the given point. (Let y be the dependent variable and let x be the independent variable)

Draw diagrams to show two vectors a and b, and the vectors a + b and a − b.
When is the magnitude of a + b less than that of a − b?
When is the magnitude of a + b equal to that of a − b?
When is |a + b| = |a| + |b|?

limx->infinity(a^1/x -1)x

Write a linear equation for the line going through the points (-2, -87) and (8, -7)

Write your equation in the form y =

Use the properties of summation to evaluate the following:

4
Σ (i^2-3i)
i=1