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Challenge problems

1. Calc I Review

A wheel of radius r = 10cm is rotating at 100 radians per minute. Attached to the rim is a steel bar of length L = 20cm, which is allowed to pivot freely. The movement of the opposite (red) end of the steel bar is constrained to the x-axis.

Calculate the speed (in cm per minute) at which the red dot it moving when the pictured angle θ is 45 degrees [Hint: you'll need a certain geometric law].

Also, verify that when the steel bar is on the x-axis, the red dot has stopped moving.

2. Section 6.1

Draw a graph of the function f(x) = 1/π e^-x on the interval [0, 1]. Find the area A(x) of the resulting shape under this curve. Then, calculate the volume V of the solid formed by revolving this shape about the x-axis.

What if instead of looking at the interval [0,1], you look at the interval [0,2]? What about the interval [0, b] where b is any positive number? Calculate the volume V(b) of the solid formed by rotating this shape about the x-axis.

Finally, calculate the limit of V(b) as b approaches infinity. In your opinion, what does this number represent?

3. Section 7.1

Suppose S(x) is a continuous, differentiable, strictly increasing function on the interval (-π/2,&pi/2) satisfying,

S''(x) = -S(x)
S(0) = 0
S'(0) = 1

Show the following, knowing nothing else about the function S:

a.

(S(x))² + (S'(x))² = 1

[Hint: Differentiate the left side to show it's constant, then find the constant]

b.

Let A(x) be the inverse function of S(x). Using part a, show that when |x|<1,

A'(x) = 1/ √1-x²

[Hint: Find A'(x) and then square it]

c.

Suppose Σ(x) is another function satisfying the exact same conditions. Show that S(x)-Σ(x) is a continuous, differentiable function whose first, second, third, etc. derivatives at x=0 are all 0. How many functions can you think of that satisfy this property? What do you think the relationship between S(x) and Σ(x) is?

4. Fun

Complete this puzzle