Physics C Mechanics

Practice Problems: Extremes and Inflections

Remember: Extreme points represent minima and maxima on motion graphs (where the first derivative is zero) and inflection points represent places on the graph where the acceleration changes sign (where the second derivative is zero). The illustration below shows the inflection points where the tangent line changes color.

1. (moderate) Graph the motion equaton given below and determine any extreme and inflection points (over the time range of -1 to +1 seconds).
x = 9t3 - 4t + 7
(with x in meters and t in seconds)

2. (moderate) Find the displacement of a particle from 4 to 8 seconds if the equation below accurately describes its motion.
v = 4t3+ 4t2 + 7t +1
(v is in m/s and t is in s)

3. (moderate) Use the graph below to determine the final position of a particle given that the initial position was at xo = 20 m.

4. (moderate) A particle is moving along a straight line according to: x = t4/6 - 7t3/6 + 3t2/2 + 5
Where t is in seconds x is in meters.
a. Find the velocity of the particle at t = 1.8 seconds.
b. Where is the particle at t = 4.1 seconds?
c. Create a v - t graph for the motion for the particle in the range from t = 0 to t = 3 seconds.

5. (moderate) v = 8t3 + 7t2/2 - 9t + 3 (with t in seconds and v in m/s)
Determine the position of the particle at t = 2.4 s if the initial position was x = 2.0 m.

6. (moderate) Extremes and inflection points have meaning in many relationships besides those found in motion studies. For example, if two variables, A and B are related by the equation A = 5B2 + 3B + 7, one can easily find the max or min for the magnitude of A by setting the first derivative dA/dB = 0. Try to use this concept to solve the following problem: You have 24 feet of fence with which to construct a rectangular rabbit enclosure. You want the enclosure to have the maximum amount of area.
a. Using calculus, determine the length of the sides of this enclosure.
b. Determine an equation that can be used to find the lengths of the sides having any length of fencing that you are given.