Using the definition of the derivative to differentiate.

Use the definition of the derivative f'(x)=  Lim  f(x+h)-f(x)     h->0  -----------                h to prove D[x+sqrtx]=1+1/2sqrtx

Finding constants to satisfy a differential equation.

Find the constants A, B, and C such that the function y=Ax^2+Bx+C satisfies the differential equation y^n+y'-2y=x^2.

Using related rates to answer a pulley question.

Two carts A and B are connected by a rope 39 feet long that passes over pulley P. The point Q is on the floor directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q? Express solution using related rate notation i.e. dy/dt or dx/dt

mu''(t) + ku(t) = 0

Suppose m and k are positive numbers. Find u so that mu''(t) + ku(t)= 0 for all numbers t and u(0)=1 and u'(0)=2. (note: u'' = second derivative of u)

Finding a Polynomial

Find a polynomial p so that: p''(t)+3p'(t) + 2p(t) = (t^2)-2 for all numbers t. (note: p''= p double prime and t^2 = t raised to the power of 2)