MATH 296 PROBLEMS 13

IGOR KRIZ

Regular problems:

1. (a) Find the derivative of

F (x) = ∫_{a}^{x^{3}} sin^{3} tdt.

(b) Find the derivative of

F (y) = ∫_{0}^{y} x sin(y)dx.

2. (a) Calculate the length of the segment of the parabola y = x^{2} between x = 0 and x = a.

(b) Calculate the length of the “spiral curve” in ℝ^{3} parametrized by

(t, cos(t), − sin(t))

for t ∈ [0, 2π].

3. Calculate the area of the hyperboloid, i.e. surface obtained by rotating the graph of y = 1/x, x ∈ [1, 2] around the x axis.

4. Calculate

lim_{n→∞} (\frac{1}{n^{2}} + \frac{1}{(n + 1)^{2}} + ... + \frac{1}{(2n)^{2}} ) .

[Express as a Riemann sum - compare to examples done in class.] Challenge problems:

5. Recall that we have shown that the Schwarzian derivative

\frac{u′′′}{u′} − \frac{3}{2} (\frac{u′′}{u′} )^{2}

is 0 for every function u = (ax + b)/(cx + d). Now suppose u is any function whose Schwartzian derivative is 0.

(a) Prove that u^{′′2}/u^{′3} is a constant function. [Differentiate.]

(b) Prove that u = (ax + b)/(cx + d). [Take the square root of the expression of (a). This is still constant. Put v = u′ and integrate the new expression dv.]

6. (a) Prove that

arctan x + arctan y = arctan (\frac{x + y}{1 − xy} ) .

[“Cheaper solution”: substitute x = tan α, y = tan β and apply tan to the formula. Prove the resulting trig. identity using the ones you know. More honest solution: Define arctan(x) = ∫_{0}^{x} \frac{dx}{1+x^{2}} and use the properies of Riemann integral.]

(b) Prove that

c arctan x = arctan (c.\frac{x}{1 − x^{c}} )

for x > 0, c > 0.