Reference no: EM132750967

Calculus Assignment -

Question 1 - The probability that Jack passes calculus II given that he passes Calculus I is 0.8. The probability that Jack passes Calculus Ii given that he fails Calculus I is 0.3. Likewise, the probability that Jack passes Calculus III given that he passes Calculus II is 0.8. The probability that Jack passes Calculus III given that he fails Calculus II is 0.3. Suppose that Jack passes Calculus I.

(a) Find the probability that Jack passes Calculus III.

(b) Find the probability that Jack passes Calculus II and Calculus III.

Question 2 - Evaluate the integral using the Fundamental Theorem of Calculus.

_π/4∫^5π/6 sin(θ) dθ =

Question 3 - Express the complex number +(1+++_l √3)+ in polar form.

Question 4 - Describe the following in detail.

(a) State De Morve's theorem.

(b) Use De Moivre's theorem to find all the cube roots of -8.

Question 5 - Let f(x) be continuous on the compact interval I, and suppose that f(x) has an infinity maximum points (i.e. points where it takes on its maximum value on I), an infinity of minimum points on I, and that between any two maximum points lies at least one minimum point.

Prove f(x) is constant on I. Why isn't sin(1/x) a counter example?

Question 6 - The cross product can also be used to find the volume of a tetrahedron. Given four points P, Q, R, S in R³, we can set u = PQ, v = PR, w = PS and calculate the volume of the parallelepiped spanned by u, v, and w as |u · (v x w)|. But how does this compare to the volume of the tetrahedron with vertices P, Q, R, S?

a) Use techniques from Calculus II to calculate the volume of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1). Compare this to the volume of the parallelepiped spanned by i, j, and k. What fraction of the parallelepiped is taken up by the tetrahedron? (You should probably draw a picture.)

b) As you will learn in Linear Algebra, this fraction is invariant under linear transformation. With this in mind, calculate the volume of the tetrahedron with vertices P = (2, -1, 3), Q = (-4, 1, 0), R = (5, 0, -2), and S = (3, 3, 3).