1. Sample Midterm 2
(1) Suppose that y = f (x) is a differentiable function on [−1, 1] so that f (sin x) = x for
−π
2 ≤ x ≤ π
2. Find f0(√ 3/2).
(2) Use implicit differentiation to find the tangent line at (2, 2) of the graph of the function 2x3− 3y2 = 4.
(3) Prove that f (x) = x − ln x is increasing for x > 1.
(4) Let y = f (x) be a function continuous on [a, b] and differentiable on (a, b). Suppose that f (a) = f (b) = 0. Show that there exists c ∈ (a, b) so that f0(c) = f (c).
(5) Compute lim
x→0+
3x + 1
x − 1
sin x
.
(6) Compute the following indefinite integrals (a)
Z
sec xdx.
(b) Z
x2ln xdx.
(c)
Z x4+ x2− 1 x3+ x dx.
(7) Sketch the graph of a rational function y = x2− 2x + 4 x − 2 . (8) A right triangle whose hypotenuse is√
3m long is revolved about one of its leg to generate a right circular cone. Find the radius, height, and volume of the cone of the greatest volume that can be made this way.
(9) Bonus
1
