Gannon Test 3 Review Problems

These flashcards cover the key concepts, definitions, and methods used in Math 125 as detailed in the lecture notes for review ahead of Test 3.

What does it mean for a function f to be increasing on an interval I?

A function f is increasing on I if for all x and y in I, if x < y then f(x) < f(y).

What does it mean for a function f to be decreasing on an interval I?

A function f is decreasing on I if for all x and y in I, if x < y then f(x) > f(y).

How do you find critical numbers for the function f(x) = 4x^2 - 6x?

Take the derivative, set it equal to zero, and solve for x.

What is the first derivative test?

A method used to determine the relative extrema of a function by analyzing the sign changes in the first derivative.

What does the extreme value theorem state?

If f is continuous on a closed interval [a, b], then f attains both a maximum and a minimum value on that interval.

Define concave up in relation to a function f on an interval I.

A function f is concave up on I if the second derivative f''(x) > 0 for all x in I.

Define concave down in relation to a function f on an interval I.

A function f is concave down on I if the second derivative f''(x) < 0 for all x in I.

What is the purpose of the second derivative test?

To determine whether a critical point is a relative maximum, relative minimum, or neither by examining the sign of the second derivative.

How do you find the points of inflection of a function?

By setting the second derivative equal to zero and solving for x to find where the concavity changes.

What is the geometric optimization problem given regarding a rectangle with a perimeter of 100 inches?

Find the length and width that maximize the area of a rectangle when the perimeter is fixed at 100 inches.