Mathematics 2A, Section D, Fall 1996

Marcel Oliver

Midterm Exam I

1. Compute the derivative of each of the following functions: (a) p(x) = a x^{100} + b,

(b) f(x) = sin^{2}(x) + cos^{2}(x),

(c) g(y) = sin(3 cos y),

(d) h(z) = \frac{a + b z}{a − b z},

where a and b do not depend on z. (5+5+10+10)

2. Consider the function

h(z) = \frac{z^{\frac{4}{3}}}{z} .

(a) Find the domain and the range of h.

(b) For which value of z is h discontinuous? Is the discontinuity removable? Explain your answer.

(c) Compute the derivative of h.

(d) For which value of z is h′ discontinuous? Is the discontinuity removable? Explain your answer.

(e) Sketch the graphs of h and h′. You may use the same coordinate

system provided you label the graphs clearly.

(5+5+5+5+10)

3. Let f(x) = \sqrt{x}.

(a) Compute f′(x) by using the power rule.

(b) Compute f′(x) by using the definition of the derivative as the limit of a difference quotient.

(10+20)

4. In the lands of Pythagoria people prefer triangular shapes over rectangular ones. So a farmer plans to enclose an isosceles triangular pasture adjacent to a straight river. He needs to enclose an area of 100 square meters, wants to use as little fence as possible, but does not know how much fence he should buy. Solve the farmer’s problem, assuming that no fencing is needed along the river. (30)