Difference Quotient
Flashcard
•
Mathematics
•
9th - 12th Grade
•
Hard
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15 questions
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1.
FLASHCARD QUESTION
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.
2.
FLASHCARD QUESTION
Front
How do you find the difference quotient for \( g(x) = 4x^2 - 2x \)?
Back
The difference quotient is \( \frac{g(x+h) - g(x)}{h} = \frac{(4(x+h)^2 - 2(x+h)) - (4x^2 - 2x)}{h} = 8x + 4h - 2 \).
3.
FLASHCARD QUESTION
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: \( \frac{h(x+h) - h(x)}{h} = \frac{\frac{1}{(x+h)-2} - \frac{1}{x-2}}{h} \).
4.
FLASHCARD QUESTION
Front
Calculate the difference quotient for \( f(x) = x^2 \) at \( x = 3 \).
Back
The difference quotient is \( \frac{f(3+h) - f(3)}{h} = \frac{(3+h)^2 - 3^2}{h} = \frac{6h + h^2}{h} = 6 + h \).
5.
FLASHCARD QUESTION
Front
What is the result of the difference quotient for \( f(x) = x^2 - 3 \) at \( x = 1 \)?
Back
The difference quotient is \( \frac{f(1+h) - f(1)}{h} = \frac{(1+h)^2 - (1^2 - 3)}{h} = \frac{2 + h}{h} = 2 + h \).
6.
FLASHCARD QUESTION
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.
7.
FLASHCARD QUESTION
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: \( \frac{\frac{1}{(x+h)-2} - \frac{1}{x-2}}{h} \).
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