precalc 1.2 flashcard A BBB

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 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.

Front

What is the standard form of a quadratic function?

Back

The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.