Calculus Answers

An example of the complete elliptic integral of the second kind, which is defined by:

E(k)=∫ π/2(upper),0(lower) (1−k^2sin^2⁡x dx), 0 ≤k ≤1.

Note that the definite integral is over the variable x, and that the result is a function of k, which is the parameter inside the integral.

(a) Consider the elliptic integral for k=0, that is E(0). Evaluate the definite integral to show that E(0)=π/2.

(b) Consider the elliptic integral for k=1, that is E(1). Evaluate the definition integral to show that E(1)=1.

(c) Now consider the elliptic integral for k=1/2, that is E(1/2). In this case we cannot find an explicit antiderivative to evalute the definite integral using the Fundamental Theorem of Calculus. We can however resort to comparison properties for definite integrals to bound the value of the integral. Starting with the fact that 0 ≤sin^(2)x ≤ 1, show that (√3)/4*π ≤ E(1/2) ≤1/2π.

Consider the function f(x)=(x−2)^3

(a) Estimate the area between the curve y=f(x), the x-axis, and the lines x=2 and x=6, using a Riemann sum with four rectangles (use the right endpoint rule).

(b) Calculate the precise area by taking the limit of a Riemann sum.

. It is well known that the rate of flow can be found by measuring the volume of blood that flows past a point in a given time period. The volume V of blood flow through the blood vessel is 2pievr dr integral from R to 0 where v = K (R^2- r^2) is the velocity of the blood through a vessel. In the velocity of the blood through a vessel, K is a constant, the maximum velocity of the blood, R is a constant, the radius of the blood vessel and r is the distance of the particular corpuscle from the center of the blood vessel.

(i) If R = 0.30 cm and v= ( 0.30- 3.33 cm/s), find the volume.

(ii) Construct and develop a general formula for the volume V of the blood flow

Consider the parabolic function ax^2+bx+c , where a is not equal to 0 , b and c are constants. For what values of a b, and c is f

(i) concave up?

(ii) concave down?

Sketch a graph that has:

1. 1 turning point, a maximum value and a y-intercept of (0,5). (3 pts)
2. End behaviours of III → I, y-intercept of 2 and 2 x-intercepts. (3 pts)
3. End behaviour of II → I, 1 x-intercept.

Plot a graph of cosx

Plot a graph of sinx

Find all numbers at which f is not continous for the given functions. Answers must be written in interval notation.

a. g(x) = x+5/x²+6x+8

b. f(x) = (3/x²) + (2/x+4) - (x/2x-3)

c. f(x) = x/(x³-x²)

d. f(x) = [(sqrtx)+10]/x

e. f(x) = 2x/x(sqrt x+4)

f. f(x) = ln srqt(x+6/4x²-9)

g. f(x) = ln srqt(x-8/x+3)

h. f(x) = [(sqrtx)+5]/4-(sqrt x)

i. f(x) = x-1/(sqrt x²-3x+2)

j. f(t) = sin^-1 (2t-3)