Find Difference Quotient & Evaluate/Approximate Limits

In summary, when finding a difference quotient, you will need to find f(x+h) and f(x), subtract them, divide by h, and then take the limit as h approaches 0. This will give you the derivative of the given function.
  • #1
Jaclbl
2
0
I understand how to find a difference quotient, but afterwards it asks me to then evaluate or approximate each limit, is that just by plugging in the given limit or is there another step?
 
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  • #2
Jaclbl said:
I understand how to find a difference quotient, but afterwards it asks me to then evaluate or approximate each limit, is that just by plugging in the given limit or is there another step?

Hi Jaclbl,

Welcome to MHB! :)

It would be much better to look at a real problem I think, but in general these limits usually require a cancellation because in the denominator you have a $\Delta x$ and the limit is $\Delta x \rightarrow 0$, so some algebraic manipulation is required to hopefully cancel something out.
 
  • #3
Ah, I don't want to do the exact problem, because it is for homework, but I'll send one of the ones that is like it, for an example.
lim F(x+h)-f(x)
h-0 h

f(x) = 4 - 2x -x2
 
  • #4
Jaclbl said:
Ah, I don't want to do the exact problem, because it is for homework, but I'll send one of the ones that is like it, for an example.
lim F(x+h)-f(x)
h-0 h

f(x) = 4 - 2x -x2

Ok, sounds good. What are $f(x+h)$ and $f(x)$ for this problem?
 
  • #5
Hello, Jaclbl!

[tex]f(x) \:=\:4-2x-x^2[/tex]

Find: [tex]\;\lim_{h\to0}\frac{f(x+h) - f(x)}{h}[/tex]

I have taught my students to make four steps.

(1) Find [tex]f(x+h)[/tex]; replace [tex]x[/tex] with [tex]x+h[/tex], and simplify.

(2) [tex]f(x+h) - f(x)[/tex]: subtract [tex]f(x)[/tex], and simplify.

(3) [tex]\frac{f(x+h)-f(x)}{h}[/tex]: divide by [tex]h[/tex] . . . factor and reduce.

(4) [tex]\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]: take limit as [tex]h\to 0.[/tex]We are given: [tex]\:f(x) \:=\:4 - 2x - x^2[/tex]

[tex](1)\;f(x+h) \:=\:4-2(x+h) - (x+h)^2 \:=\:4 - 2x - 2h - x^- 2xh - h^2[/tex]

[tex](2)\;f(x+h)- f(x)[/tex]
. . . [tex]=\: (4-2x-2h - x^2 - 2xh - h^2)-(4-2x-x^2) \:=\:\text{-}2h -2xh - h^2[/tex]

[tex](3)\;\frac{f(x+h)-f(x)}{h} \;=\;\frac{\text{-}2h-2xh-h^2}{h} \;=\;\frac{h(\text{-}2-2x-h)}{h} \;=\;\text{-}2-2x-h[/tex]

[tex](4)\;\lim_{h\to0}\frac{f(x+h)-f(x)}{h} \;=\;\lim_{h\to0}(\text{-}2-2x-h) \;=\;\text{-}2-2x[/tex]Therefore, the derivative of [tex]f(x)\:=\:4-2x-x^2[/tex] is: [tex]\;f'(x) \:=\:-2-2x[/tex]
 

Related to Find Difference Quotient & Evaluate/Approximate Limits

1. What is the difference quotient?

The difference quotient is a mathematical expression used to find the slope of a curve or the rate of change of a function at a specific point. It is calculated by taking the difference between the values of the function at two points and dividing it by the difference between the x-values of those points.

2. How do I find the difference quotient?

To find the difference quotient, you need to first choose two points on the function, (x1, y1) and (x2, y2), where x2 is not equal to x1. Then, plug these values into the difference quotient formula (f(x2) - f(x1)) / (x2 - x1) to get the slope or rate of change at that point.

3. What is the purpose of finding the difference quotient?

The difference quotient is used to approximate the slope or rate of change of a function at a specific point. It is also used in finding the derivative of a function, which is an important concept in calculus and is used to model real-world situations such as motion and growth.

4. How do I evaluate/approximate limits using the difference quotient?

To evaluate or approximate limits using the difference quotient, you first need to find the difference quotient for a given function. Then, as the difference between the two x-values gets smaller and smaller, the value of the difference quotient will approach the value of the limit at that point. This process is known as taking the limit of the difference quotient.

5. Can the difference quotient be used for all types of functions?

Yes, the difference quotient can be used for all types of functions, including linear, quadratic, exponential, and trigonometric functions. However, it may be more difficult to calculate for more complex functions and may require the use of limit laws and other calculus techniques.

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