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.

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).

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

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

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) = 1− 4/x², x ≠ 0