Skill 16 - Difference Quotient

Front

What is the definition of the difference quotient?

Back

The difference quotient is defined as \( \frac{f(a+h) - f(a)}{h} \), which represents the average rate of change of the function \( f \) over the interval from \( a \) to \( a+h \).

Front

How do you calculate the average rate of change of a function?

Back

To calculate the average rate of change of a function \( f \) over an interval \( [a, b] \), use the formula \( \frac{f(b) - f(a)}{b - a} \).

Front

What is the average rate of change of \( f(x) = x^2 - x + 1 \) from \( a \) to \( a+h \)?

Back

The average rate of change is given by \( \frac{(a+h)^2 - (a+h) + 1 - (a^2 - a + 1)}{h} = 2a - 1 + h \).

Front

Find the average rate of change of \( f(x) = x^2 - 3x + 1 \) on the interval \([-1, 3]\).

Back

The average rate of change is \( \frac{f(3) - f(-1)}{3 - (-1)} = \frac{(-1) - 4}{4} = -1 \).

Front

What does the term 'average rate of change' signify in a function?

Back

The average rate of change signifies how much the function's output changes, on average, for a unit change in the input over a specified interval.

Front

What is the significance of the limit of the difference quotient as \( h \) approaches 0?

Back

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

Front

What is the derivative of \( f(x) = x^2 \)?

Back

The derivative is \( f'(x) = 2x \), which represents the instantaneous rate of change of the function at any point \( x \).

Front

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

Back

To find the slope of the tangent line at a point \( a \), calculate the derivative \( f'(a) \).

Front

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

Back

The difference quotient is a foundational concept for understanding derivatives; the derivative is the limit of the difference quotient as \( h \) approaches 0.

Front

What is the average rate of change of \( f(x) = 2x + 3 \) from \( x = 1 \) to \( x = 4 \)?

Back

The average rate of change is \( \frac{f(4) - f(1)}{4 - 1} = \frac{(2(4) + 3) - (2(1) + 3)}{3} = \frac{8 - 2}{3} = 2 \).