COPY: AP Precalculus Unit 1 Review

Front

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

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval where x1 < x2, the value of the function at x1 is less than the value at x2 (f(x1) < f(x2)).

Front

Define concavity in the context of a graph of a function.

Back

Concavity refers to the direction in which a graph curves. A graph is concave up if it opens upwards (like a cup) and concave down if it opens downwards (like a frown).

Front

What is the average rate of change of a function?

Back

The average rate of change of a function f over an interval [a, b] is given by (f(b) - f(a)) / (b - a). It represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

Front

How can you determine if a function is concave up or concave down using the second derivative?

Back

If the second derivative of a function f, denoted f''(x), is positive on an interval, the function is concave up on that interval. If f''(x) is negative, the function is concave down.

Front

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

Back

A function is decreasing on an interval if, for any two points x1 and x2 in that interval where x1 < x2, the value of the function at x1 is greater than the value at x2 (f(x1) > f(x2)).

Front

Explain the significance of the x-intercepts of a function.

Back

The x-intercepts of a function are the points where the graph crosses the x-axis. These points occur when the function value is zero (f(x) = 0). They are important for understanding the roots of the function.

Front

What is a polynomial function?

Back

A polynomial function is a function that can be expressed in the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

Front

Define the term 'interval' in mathematics.

Back

An interval is a set of real numbers that contains all numbers between any two numbers in the set. Intervals can be open (not including endpoints) or closed (including endpoints).

Front

What does it mean for a function to have a local maximum?

Back

A function has a local maximum at a point x = c if f(c) is greater than or equal to the values of the function at points near c. It is the highest point in a small neighborhood around c.

Front

What is the difference between a function being increasing and concave up?

Back

A function being increasing means that its output values rise as the input values increase. Concavity refers to the curvature of the graph; a concave up function has a shape that opens upwards, indicating that the slope of the function is increasing.