Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
πcos πx

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

g(x) = −x² − 6x − 9,−4 ≤ x < ∞

53. Distance between two ships At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship B was sailing east at 8 knots and continued to do so all day.

a. Start counting time with t=0 at noon and express the distance s between the ships as a function of t.

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

Finding displacement from an antiderivative of velocity

a. Suppose that the velocity of a body moving along the s-axis is

ds/dt = v = 9.8t − 3.

i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫tanθ sec²θ dθ = sec³θ / 3 + C

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = (x − 1)(x + 2)(x − 3)