PRACTICE MIDTERM 2-
Problem 1- (i) Complete the sentence: "A function f is continuous at x = a if . . ." Give an example of a function f and some a in the domain of f such that f is not continuous at x = a.
(ii) Complete the sentence: "A function f is differentiable at x = a if . . ." Give an example of a function f and some a in the domain of f such that f is not differentiable at x = a.
Problem 2- (i) Find the equation of the tangent line to the curve y = (2 ln x/x3) - (sin-1(x))5 at the point where x = 1/2.
(ii) Find the tangent line to the graph x2 + 2y4 = 3 at the point (1, 1).
Problem 3- (i) Let f(x) be a function of x such that (f(x))x = 1. Find f'(x) in terms of f.
(ii) Let f, g be differentiable functions such that g(x) > 0 for all x. Set h(x) = f(xg(x)). Find h' in terms of f, g, f', g'.
(iii) Let f(x) = 3x. Find a function F whose derivative is f.
Problem 4- Strontium-90 has a half-life of 28 days. A sample has a mass of 50 mg initially.
(a) How much of the sample remains after t days?
(b) How long does it take the sample to decay to a mass of 2 mg?
Problem 5- (i) State Rolle's Theorem.
(ii) Use Rolle's Theorem to prove the Mean Value Theorem.
(iii) Let f(x) = (1/5)x5 - (2/3)x3 + x. Prove that f(π) > f(1).
Problem 6- Find limx→0^+ x ln x.
Problem 7- A pitcher at P = (0, 2) throws a ball whose position at time t is B(t) = (x(t), y(t)) where x(t) = t and y(t) = -(1/32) (x(t))2 + 2. You are watching the ball from the point A = (8, 2). Let θ(t) be the angle ∠PAB(t). How fast (in radians) is θ(t) changing when the ball hits home plate, i.e. when B(t) = (8, 0)?
Problem 8- Consider the graph of the function f(x) = x3 - x.
(a) Find the regions of the graph where f(x) is increasing. Find the regions of the graph where f(x) is decreasing. Find all local extrema of f. Which of these are global extrema?
(b) Find the regions of the graph where f(x) is concave up. Find the regions of the graph where f(x) is concave down. Find all inflection points.