precalc 1.2 flashcard B AAA

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 represents the average rate of change of the function over the interval [x, x+h].

Front

Find 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

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

How do you find the derivative of f(x) = x^2?

Back

Using the power rule, the derivative f'(x) = 2x.

Front

What is the power rule in differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Front

What is the significance of the slope of the tangent line to a curve at a point?

Back

The slope of the tangent line at a point on a curve represents the instantaneous rate of change of the function at that point.

Front

What is the formula for the slope of a line?

Back

The slope (m) of a line through two points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1).

Front

What is the difference between average rate of change and instantaneous rate of change?

Back

Average rate of change is calculated over an interval, while instantaneous rate of change is calculated at a specific point.

Front

What is the limit in calculus?

Back

A limit is a value that a function approaches as the input approaches some value.