precalc 1.2 flashcard A

Front

What is the difference quotient for a function f(x)?

Back

The difference quotient is defined as (f(x + h) - f(x)) / h, which measures the average rate of change of the function over the interval [x, x + h].

Front

Calculate the difference quotient for f(x) = x^2 - x + 1.

Back

The difference quotient is (f(x + h) - f(x)) / h = ((x + h)^2 - (x + h) + 1 - (x^2 - x + 1)) / h = (2x + h - 1) = 2x - 1 + h.

Front

What does the term 'average rate of change' refer to in calculus?

Back

The average rate of change of a function over an interval [a, b] is given by (f(b) - f(a)) / (b - a).

Front

Define the term 'function' in mathematics.

Back

A function is a relation that assigns exactly one output for each input from a given set, typically denoted as f(x).

Front

What is the significance of the derivative in calculus?

Back

The derivative represents the instantaneous rate of change of a function at a given point and is defined as the limit of the difference quotient as h approaches 0.

Front

Explain the concept of limits in calculus.

Back

A limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value, often used to define derivatives and integrals.

Front

What is the formula for the derivative of a polynomial function?

Back

The derivative of a polynomial function f(x) = ax^n is f'(x) = n * ax^(n-1).

Front

How do you find the slope of a tangent line to a curve at a point?

Back

The slope of the tangent line at a point on a curve is found by calculating the derivative of the function at that point.

Front

What is the difference between a function and a relation?

Back

A function is a specific type of relation where each input is associated with exactly one output, while a relation can have multiple outputs for a single input.

Front

What is the role of 'h' in the difference quotient?

Back

In the difference quotient, 'h' represents a small change in the input value x, used to calculate the average rate of change.