precalc 1.2 flashcard B

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].

Tags

CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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.

Tags

CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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).

Tags

CCSS.HSF.IF.B.6

CCSS.8.F.B.4

Front

What is the derivative of a function?

Back

The derivative of a function at a point is the limit of the difference quotient as h approaches 0, representing the instantaneous rate of change.

Front

Define the term 'function'.

Back

A function is a relation that assigns exactly one output for each input from a set called the domain.

Tags

CCSS.HSF.IF.A.1

CCSS.8.F.A.1

Front

What is the significance of the variable 'h' in the difference quotient?

Back

The variable 'h' represents a small change in the input value, used to calculate the average rate of change.

Tags

CCSS.HSF.IF.B.6

CCSS.8.F.B.4

Front

Explain the concept of limits in calculus.

Back

A limit is a value that a function approaches as the input approaches some value. It is fundamental in defining derivatives and integrals.

Front

What is the formula for the difference quotient?

Back

The formula for the difference quotient is (f(x + h) - f(x)) / h.

Tags

CCSS.HSF.IF.B.6

CCSS.8.F.B.4

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 is found by calculating the derivative of the function at that point.

Front

What is the relationship between the difference quotient and the derivative?

Back

The derivative is the limit of the difference quotient as h approaches 0.