Math 16a worksheet 12 let f and g be two differentiable

Question

MATH 16A WORKSHEET 12-

(1) The mass of a sample of radioactive material is described by the function f(t), which satisfies the differential equation

F'(t) = (ln 6)f(t)

and satisfies f(1) = 24. Find the formula for f(t). What is f(0)? At what time is f(t) = 144?

(2) Let f and g be two differentiable functions. Find an anti-derivative (in terms of f and g) for each of the following expressions.

(a) f'(x)g(x) + f(x)g'(x)

(b) f'(g(x)) · g'(x)

(c) e^f(x)f'(x)

(d) f'(t)/f(t)

(e) 2g(x)g'(x)

(f) x⁴

(g) (1/x) + x + e^3x

(3) Find all functions f(t) such that f'(t) = (-1/x²) + x² and f(1) = 0.