Precalculus requires a thorough understanding of functions and the language used to work with functions.
The lessons below will give you the required review.
Be sure to click-click-click through some of the exercises in each of these lessons!
The lessons will open in a new tab/window.
Then, do the lesson on this page to practice with difference quotients, which are particularly important in calculus.
Consider a point $\,\bigl(x,f(x)\bigr)\,$ on the graph of a function $\,f\,$.
If $\,h\,$ is a small positive number, then $\,x+h\,$ lies a little to the right of $\,x\,$.
If $\,h\,$ is a small negative number, then $\,x+h\,$ lies a little to the left of $\,x\,$.
In both cases, $\,\bigl(x+h,f(x+h)\bigr)\,$ is a point on the graph of $\,f\,$, likely close to the original point $\,(x,f(x))\,$.
The slope of the line through $\,\bigl(x,f(x)\bigr)\,$ and $\,\bigl(x+h,f(x+h)\bigr)\,$ is: $$ \cssId{s13}{\text{slope}} \cssId{s14}{= \frac{\text{change in $y$}}{\text{change in $x$}}} \cssId{s15}{= \frac{f(x+h) - f(x)}{(x+h)-x}} \cssId{s16}{= \frac{f(x+h)-f(x)}{h}} $$
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closer to the original point $\,\bigl(x,f(x)\bigr)\,$, To keep the graph uncluttered, the labels for the new (closer) points $\bigl(x+h,f(x+h)\bigr)$ are not shown. |
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Recall that the result of a division problem, $\,\frac{a}{b}\,$, is called a quotient,
and the result of a subtraction problem, $\,a - b\,$, is called a difference.
The expression $\displaystyle\,\frac{f(x+h)-f(x)}{h}\,$ is called a difference quotient,
because it is a quotient of differences:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
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