A few quick questions with derivatives using limit def.

In summary, the first one says that you can use the limit definition to find the derivative of the function. The second one says that you can use the limit definition to find the derivative at the indicated point. The last one says that you can use the limit process to find the slope of the graph of the function at the specified point.
  • #1
Precal_Chris
40
0

Homework Statement


The first one says use the limit definition to find the derivative of the function.
F(x)= 1/(2x-4)

the second one is use the limit definition to find the derivative at the indicated point..
f(x)= -x^3 + 4x^2, at (-1,5)

the last one is use the limit process to find the slope of the graph of the function at the specified point.
f(x)= sqrt of (x + 10), at (-1,3)


Homework Equations


i know that this equation is used to help find it..
F(x + P)- f(x)/ P
p=the change in x..or delta x



The Attempt at a Solution


i don't even know where to begin ...but any help at all is welcomed and thanked.
 
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  • #2
Precal_Chris said:

Homework Statement


The first one says use the limit definition to find the derivative of the function.
F(x)= 1/(2x-4)

the second one is use the limit definition to find the derivative at the indicated point..
f(x)= -x^3 + 4x^2, at (-1,5)

the last one is use the limit process to find the slope of the graph of the function at the specified point.
f(x)= sqrt of (x + 10), at (-1,3)


Homework Equations


i know that this equation is used to help find it..
F(x + P)- f(x)/ P
p=the change in x..or delta x



The Attempt at a Solution


i don't even know where to begin ...but any help at all is welcomed and thanked.

Well, first of all the definition of the derivative, as you may know is:

[tex]f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}[/tex]


For your first function it would be

[tex]f'(x)=\lim_{h\rightarrow 0}\frac{\frac{1}{2(x+h)-4}-\frac{1}{2x-4}}{h}[/tex] Now you can go from here, right?

Second one:

[tex]f'(-1)=\lim_{x\rightarrow -1}\frac{-x^{3}+4x^{2} -f(-1)}{x-(-1)}[/tex]

Third one

[tex]slope=m=f'(-1)=\lim_{x\rightarrow -1}\frac{\sqrt{x+10}-\sqrt{-1+10}}{x-(-1)}[/tex]

Now all you need to do is evaluate those limits. DO u know how to do it?
 
  • #3
Blah, i forgot: You NEED to show your work next time, before anyone here can help you.
 
  • #4
For example, if I wanted to find the derivative of the function 2x^2 + 5 using the definition of a limit, I would use the formula:

[tex]\lim_{\substack{h\rightarrow 0}}\frac{f(x+h)-f(x)}{h}=\lim_{\substack{h\rightarrow 0}}\frac{2(x+h)^2+5-(2x^2+5)}{h}[/tex]


[tex]=\lim_{\substack{h\rightarrow 0}}\frac{2x^2+4xh+2h^2+5-2x^2-5}{h}=\lim_{\substack{h\rightarrow 0}}\frac{4xh+2h^2}{h}[/tex]


[tex]=\lim_{\substack{h\rightarrow 0}}4x+2h=4x[/tex]

See how f(x) = 2x^2 + 5 and f(x+h) = 2(x+h)^2 + 5?
 
Last edited:
  • #5
To the OP:In order to receive further help, please show us what you have tried so far, and point out where are u stuck. Remember you are supposed to do your own homework not us.!
 
  • #6
ok i didnt know that
 

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.

2. How is a derivative calculated using the limit definition?

The limit definition of a derivative involves taking the limit of the difference quotient, where the difference between two points on the function is divided by the difference between their x-coordinates. As the distance between the two points approaches zero, the resulting value is the derivative at that point.

3. Why is the limit definition used to calculate derivatives?

The limit definition is used because it provides a general method for calculating derivatives of any function, including non-polynomial functions. It also allows for the calculation of derivatives at specific points rather than just an average rate of change over an interval.

4. Can the limit definition of a derivative be used for all types of functions?

Yes, the limit definition of a derivative can be used for all types of functions, including polynomial, trigonometric, exponential, and logarithmic functions. However, for some functions, it may be more efficient to use other methods to calculate derivatives.

5. What is the relationship between derivatives and tangents?

The derivative of a function at a specific point represents the slope of the tangent line to the function at that point. This means that the tangent line is the line that best approximates the behavior of the function at that point, and the derivative is the rate of change of the function along that line.

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