assignments

Assignment 1

• Differentiate the following functions:
  • \( f_1(x) = \sin^5 \left( x + \frac{1}{x} \right) \)
  • \( f_2(x) = e^4x \cos(x^4) \)
  • \( f_3(x) = x^x \)
  • \( f_4(x) = \cos^2 ( \sin^2 (x) ) \)

• Evaluate the following integrals:
  • \( \int \sin^3(x)\cos^2(x) \ dx \)
  • \( \int x^2e^{-x} \ dx \)
  • \( \int \frac{5x^3 - 3x^2 + 2x - 1}{x^4 + x^2} \ dx \)
  • \( \int \frac{1}{(4x^2 + 9)^2} \ dx \)
  • \( \int x^2 \sqrt{x^3 + 9} \ dx \)

• Show that \( \int_0^1 (1-x^2)^n \ dx = \frac{2^{2n}(n!)^2}{(2n+1)!} \). [Hint: if \( I_n \) denotes this integral, can you express \( I_{k+1} \) in terms of \( I_{k} \)?]

Assignment 2

• Show that the triangle with vertices \( (-2,4,0), (1,2,-1), (-1,1,2) \) is equilateral.
• Find all values of \( x \) for which the vectors \( ( -6,x,2) \) and \( (x,x^2,x) \) are orthogonal.
• Find two unit vectors, each orthogonal to both \( (1,1,0) \) and \( (1,-1,1) \).
• Find the equation of the plane that passes through the point \( (1,2,3) \) and contains the line \( x=3t\), \( y=1+t\), \(z=2-t\).
• Find the surface area generated by rotating a loop of the curve \( 8y^2 = x^2(1-x^2) \) about the \(x\)-axis.

Assignment 3

• Find the limits or show that they do not exist:
  • \( \lim_{(x,y) \rightarrow (0,0)} \frac{y^4}{x^4 + 3y^4} \)
  • \( \lim_{(x,y) \rightarrow (2,1)} \frac{4-xy}{x^2 + 3y^2} \)
• Let \( f(x,y) = e^{xy^2} \). Find \( f_{xxx}(x,y) \).
• The ellipsoid \( 4x^2 + 2y^2 + z^2 = 16 \) intersects the plane \(y=2\) in an ellipse. Find parametric equations for the tangent line to this ellipse at the point \( (1,2,2) \).
• The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a rate of 2.5 in/s. At what rate is the volume of the cone changing when the radius is 120 in and the height is 140 in?
• Let \( f(x,y) = xe^{-y} + 3y \). Find the maximum rate of change of \(f\) at the point \( (1,0) \) and the direction in which it occurs.
• Show that every plane that is tangent to the cone \( x^2 + y^2 = z^2 \) passes through the origin.
• Find three positive numbers whose sum is 100 and whose product is a maximum. More generally, find three positive numbers whose sum is 100 such that \( x^ay^bz^c\) is a maximum.
• Find the radius of convergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n^25^n}\)