1. Explain why or why not. Determine whether the following statements are true and give an explanation or a counterexample:

a) The function sec x is not differentiable at x= π/2

b) If the acceleration of an object remains constant then, its velocity is constant.

c) If the acceleration of an object moving along a line is always 0 then its velocity is constant.

d) It is impossible for the instantaneous velocity at all times a ≤ t ≤ b to equal the average velocity over the interval a ≤ t ≤ b.

e) a moving object can have a negative acceleration and increasing speed. 

f) The function x sin x can be differentiated without using the chain rule.

g) The function e^√(x+1) must be differentiated using chain rule.

h) The derivative of a product is not the product of the derivatives, but the derivative of a composition is a product of derivatives.

i) d/dx P(Q(X)) = P' (X) Q'(X) 

2.   500-liter (L) tank is filled with pure water. At time t = 0, a salt solution begins flowing into the tank at a rate of 5L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t ≥ 0 is given by:

M(t) = 250(1000-t)(1-10^-30 (1000-t)¹⁰ )

and the volume of solution in the tank is given by

V(t) = 500-0.5t 

a. Graph the mass function and verify that M(0)=0

b. Graph the volume function and verify that the tank is empty when t=1000 min

c. The concentration of the salt solution in the tank (in g/L) is given by C(t)=M(t)/V(t). Graph the concentration function and comment on its properties. 

Specifically, what are C(0) and lim t→1000-  C(t) ?

d) Find the rate of change of the mass M ' (t), for 0 ≤ t ≤ 1000.

e) Find the rate of change of the concentration C ' (t), for 0 ≤  t ≤ 1000.

f) For what times is the concentration solution increasing? Decreasing?