Questions from General Calculus

Q: Differentiate. f (x) − x2 sin x

Differentiate. f (x) − x2 sin x

Q: Suppose f is differentiable on R and α  is a real

Suppose f is differentiable on R and α  is a real number. Let F(x) = f (ex) and G(x) = [f(x)]α. Find expressions for (a) F’ (x) and (b) G’ (x).

Q: Suppose f is differentiable on R. Let F(x)

Suppose f is differentiable on R. Let F(x) − f (ex) and G(x) = ef (x). Find expressions for (a) F ‘(x) and (b) G ‘(x).

Q: Find equations of the tangent lines to the curve y= ln

Find equations of the tangent lines to the curve y= ln x/x at the points (1, 0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines.

Q: Let g(x) = ecx + f (x)

Let g(x) = ecx + f (x) and h(x) = ekx f (x), where f (0) = 3, f ‘(0) = 5, and f ‘’(0) = 22. (a) Find g ‘(0) and g ‘‘(0) in terms of c. (b) In terms of k, find an equation of the tangent line to the gr...

Q: Let r(x) = f (g(h(

Let r(x) = f (g(h(x))) , where h(1) = 2, g(2) = 3, h’(1) = 4, g’(2) = 5, and f ‘(3) = 6. Find r’(1).

Q: If g is a twice differentiable function and f (x)

If g is a twice differentiable function and f (x) = xg(x2), find f ‘’ in terms of g, g’, and g’’.

Q: If F(x) = f (3 f(4

If F(x) = f (3 f(4 f (x))), where f (0) = 0 and f ‘(0) = 2, find F’(0).

Q: If F(x) = f (x f (x

If F(x) = f (x f (x f (x))), where f (1) = 2, f (2) = 3, f ‘(1) = 4, f ‘(2) = 5, and f ‘(3) = 6, find f’(1).

Q: Show that the function y=e2x (A cos 3x +

Show that the function y=e2x (A cos 3x + B sin 3x) satisfies the differential equation y’’ - 4y’ + 13y=0.