Calculus Answers

If P(x) =√x, show that P(x+h) - P(x) = h(√x+h + √x).

If Φ (r)= 2^r, show that Φ (r+1) =2 Φ(r).

If F(z)= log z, show that F(xy) = F(x) +F(y).

Using the methods suggested in the preceding problem, find the area of the trapezoid bounded by the line y= x+3, the ordinates x=1, x=3, and the x axis.

A ball thrown straight up is located s feet above the ground at t seconds after it is thrown in accordance with the formula s= 112t-16 t^2. Find a formula for the velocity of the ball and find (a) the time required to reach its highest point, (b) the distance of the highest point above the ground, and (c) the acceleration of the ball at this point.

A ball rolling down an incline travels s feet in t seconds, where s= 5t^2. Derive a formula for the velocity of the ball at time t= t0. How fast is it going (a) after 2 seconds (b) after it has rolled 80 feet?

Find the derivative of the function f(x)=√x with respect to x at x0.

Find the rate of change of the function f(x)= x^3 with respect to x at x0.

1. Write

(2−i) (4−3i)2

3+2i

(4)

in the form a+bi, where a,b ∈ R.

Evalu‎ate ∫_C (x + 2y) ds, w‎here C is the cu‎rve defi‎ned by y = √(4 − x²), for x ∈ [0, 1].