1 Find the following derivatives:

 

a)

 

b)

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2. Evaluate

 

a)

 

b)

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3.Using the fact that y = arcsin(x) if and only if x = sin(y), along with implicit differentiation, prove that

 

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  1. Find

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5. Assume that f(x) and g(x) are differentiable in an open interval containing a. Suppose that f(a) = g(a) = 0 and g(a) 0. Prove L'Hospital's Rule:

 

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6. Let P(t) be the population of rabbits in an area, with t measured in months. Suppose that 2 months after t = 0 there are 2000 rabbits in the field and 5 months after t = 0 there are 5000 rabbits. How many rabbits were there originally, i.e., at t = 0? Assume that the number of rabbits is growing exponentially.

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7a. Prove, for any positive integer n, that

 

b. Find

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8. Let

 

for x in the domain D = .

 

  1. Does f'(x) always have the same sign? Why? If so, is it positive or negative?

 

  1. Does f have an inverse function on D? Why or why not?

 

c. Find , and then find

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