I have never seen any examples of these problems, and they showed up on a review sheet for the final exam in May. If you can help me through these or explain to me what I should do for each step it would help me a lot. Thanks.

1. A particle moves along a line so that at any time (t) its position is given by x(t)=πt+cos(πt). btw, π is pi.

a) Find the velocity at time t.

b) Find the acceleration at time t.

c) What are all the values of t, 0≤t≤3, for which the particle is at rest?

d) What is the maximum velocity over the interval 0≤t≤3?

2. Let f be the function defined by f(x)=(x^2 +1)e^-x for all x such that -4≤x≤4.

a) For what value of x does f reach its absolute maximum? Justify.

b) Find the x-coordinates of all points of inflection of f. Justify.

3. Let p and q be real numbers and let f be the function defined by:

f(x)={1+3p(x-2)+(x-2)^2 for x≤2

f(x)={qx+p for x>2

a) Find the value of q, in terms of p, for which f is continuous at x=2.

b) Find the values of p and q for which f is differentiable at x=2.

c) If p and q have the values determined in part b, is f" a continuous function? Justify.