Solve Derivative Problem: Lim {P(x+3h)+P(x-3h)-2P(x)}/h^2

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In summary, the conversation is about finding the limit of a polynomial function using Taylor-series expansions and L'Hopital's rule. The correct answer is D) 9P''(x) and the process involves differentiating the function twice and setting h=0.
  • #1
lhuyvn
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Hi members,

Could anyone help me with the problem following?

If x is a real number and P is a polynomial function, then


lim {P(x+3h)+P(x-3h)-2P(x)}/h^2
h->0


A)0
B)6P'(x)
C)3P''(x)
D)9P''(x)
E) 00

I guess D should be the answer, I need an explanation.

Thank You
 
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  • #2
The easiest way is to use Taylor-series expansions of the terms P(x+3h), P(x-3h)
to verify your guess.
Then we have, for example:
P(x+3h)=P(x)+P'(x)3h+1/2P''(x)(3h)^2+O(h^3)

O(h^3) is a higher order term, i.e lim h->0 O(h^3)/h^2=0
 
  • #3
Thank you, arildno

It couldn't be more wonderful solutions.
 
  • #4
It looks messy, but it's really no different from any other limit problem. What is typically the easiest way to find the limit of 0/0? L'Hopital's rule!

Differentiate top and bottom with respect to h (not x!) twice, using chain rule for top terms. So the first differentiation gives [3P'(x+3h)-3P'(x-3h)]/2h (notice the third term has no h, so drops out). The second round, you get [9P''(x+3h)+9P''(x-3h)]/2. Then setting h=0 gives the desired answer.
 
  • #5


Hi there,

The correct answer for this derivative problem is D) 9P''(x). This can be explained using the definition of a derivative. The limit of a derivative as h approaches 0 is equal to the derivative of the function at that point. In this case, the function is P(x) and the point is x. Therefore, the limit can be rewritten as:

lim {P(x+h) - 2P(x) + P(x-h)}/h^2

Using the definition of a derivative, we can expand the polynomial P(x+h) and P(x-h) as:

lim {[P(x) + P'(x)h + 1/2P''(x)h^2 + ...] - 2P(x) + [P(x) - P'(x)h + 1/2P''(x)h^2 - ...]}/h^2

Simplifying and canceling out the terms with h^2, we are left with:

lim {2P''(x)h^2}/h^2

As h approaches 0, h^2 also approaches 0, leaving us with just 2P''(x). However, the problem asks for the limit as h^2, so we need to multiply by 9 to get the correct answer of 9P''(x).

I hope this helps explain the solution to this derivative problem. Let me know if you have any further questions. Good luck!
 

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function with respect to its independent variable. It is commonly denoted as f'(x) or dy/dx.

2. How do you solve derivative problems?

To solve derivative problems, you need to first identify the function and its independent variable. Then, use the rules of differentiation such as the power rule, product rule, quotient rule, and chain rule to find the derivative. Finally, plug in the given values to solve for the derivative.

3. What is the limit in a derivative problem?

The limit in a derivative problem represents the value that a function approaches as its independent variable approaches a certain point. It is denoted as lim and is used to find the derivative at a specific point.

4. What is the meaning of h in the derivative problem formula?

The variable h in the derivative problem formula represents the change in the independent variable. It is used to calculate the slope of the line tangent to the function at a specific point.

5. How do you solve the given derivative problem?

To solve the given derivative problem, first plug in the given values for x and h into the given formula. Then, use algebraic manipulation and the rules of differentiation to simplify the expression. Finally, take the limit as h approaches 0 to find the derivative at the given point.

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