Difference Quotient

Front

What is the difference quotient?

Back

The difference quotient is a formula that gives the average rate of change of a function over an interval. It is defined as \( \frac{f(x+h) - f(x)}{h} \) where \( h \) is the change in the input value.

Front

How do you find the difference quotient for \( g(x) = 4x^2 - 2x \)?

Back

The difference quotient is \( \frac{g(x+h) - g(x)}{h} = \frac{(4(x+h)^2 - 2(x+h)) - (4x^2 - 2x)}{h} = 8x + 4h - 2 \).

Front

What is the first step in finding the difference quotient for \( h(x) = \frac{1}{x-2} \)?

Back

The first step is to set up the difference quotient: \( \frac{h(x+h) - h(x)}{h} = \frac{\frac{1}{(x+h)-2} - \frac{1}{x-2}}{h} \).

Front

Calculate the difference quotient for \( f(x) = x^2 \) at \( x = 3 \).

Back

The difference quotient is \( \frac{f(3+h) - f(3)}{h} = \frac{(3+h)^2 - 3^2}{h} = \frac{6h + h^2}{h} = 6 + h \).

Front

What is the result of the difference quotient for \( f(x) = x^2 - 3 \) at \( x = 1 \)?

Back

The difference quotient is \( \frac{f(1+h) - f(1)}{h} = \frac{(1+h)^2 - (1^2 - 3)}{h} = \frac{2 + h}{h} = 2 + h \).

Front

What does the difference quotient represent in calculus?

Back

The difference quotient represents the slope of the secant line between two points on the graph of a function, and as \( h \) approaches 0, it approaches the derivative of the function.

Front

How do you simplify the difference quotient for \( h(x) = \frac{1}{x-2} \)?

Back

To simplify, substitute \( h(x+h) \) and \( h(x) \) into the difference quotient and combine the fractions: \( \frac{\frac{1}{(x+h)-2} - \frac{1}{x-2}}{h} \).

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, and it is used to calculate the average rate of change of the function over that interval.

Front

What is the formula for the difference quotient?

Back

The formula for the difference quotient is \( \frac{f(x+h) - f(x)}{h} \).

Front

How do you find the difference quotient for a linear function?

Back

For a linear function \( f(x) = mx + b \), the difference quotient simplifies to \( m \), the slope of the line, since the change in output is constant.