Help with this differential calculus

[GaussianSurface]

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Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order to ask for help, hope you can help me.
Down below is the proble and the answer I've madeSuppose that f is given for x in the interval [0,12] by

x= 0 2 4 6 8 10 12
f(x)= -13 -16 -17 -16 -14 -11 -8
A. Estimate f'(2) using the values of f in the table.

Use two decimals.

f'(2) is approximately -16 (my answer)

B. For what values of x does f'(x) appear to be positive?

(Give your answer as an interval. Use integers.)

( -infinity, +infinity) (my answer)

C. For what values of x does f'(x) appear to be negative?

( , ) (Withouth answer yet)

 

[Svein]

For this type of problems, the best way is usually to calculate the divided differences. Since the differences in the x values are constant (2), your first divided difference will be (-16 - (-13))/2 = -3/2. Calculate those for all f values.

What you have calculated is the secant between successive values of (x, f(x)). A secant is an approximation to a tangent. Since you need the derivative at 2, you need to inspect both secants involving 2.

As to "infinity" - for what values of x is f(x) defined?

 

[beastforever]

i'm getting f'(2) to be -1.00 by taking the average of the 2 secants..

 

[GaussianSurface]

For this type of problems, the best way is usually to calculate the divided differences. Since the differences in the x values are constant (2), your first divided difference will be (-16 - (-13))/2 = -3/2. Calculate those for all f values.

What you have calculated is the secant between successive values of (x, f(x)). A secant is an approximation to a tangent. Since you need the derivative at 2, you need to inspect both secants involving 2.

As to "infinity" - for what values of x is f(x) defined?

Let me see if I got you said:
then basing on what you said I'd got dividing the differences
=> -3/2 first value for f
=> -1/4 second value for f
=>-1/6 third ''
=> 2/8 = 1/4fourth
=> 3/10 fifth
=> 3/12 = 1/3 sixth
what you said about the infinity I didn't understand what you meant I'm kind of lost on those last two problems, help please.

 

[beastforever]

Let me see if I got you said:
then basing on what you said I'd got dividing the differences
=> -3/2 first value for f
=> -1/4 second value for f
=>-1/6 third ''
=> 2/8 = 1/4fourth
=> 3/10 fifth
=> 3/12 = 1/3 sixth
what you said about the infinity I didn't understand what you meant I'm kind of lost on those last two problems, help please.

I think what he/she meant there was to take the values of secants with a value before 2 (0) and one after 2(4) and then approximate the value of the tangent.. and as of infinity, I think he/she wanted to ask you the values for which we know f(x), because, according to the information here, we have values of f(x) when x is an even integer, while we don't know the behavior of f(x) when x is an odd integer, and also we don't know the behavior for real numbers in general.. hope that was helpful.

 

[GaussianSurface]

I think what he/she meant there was to take the values of secants with a value before 2 (0) and one after 2(4) and then approximate the value of the tangent.. and as of infinity, I think he/she wanted to ask you the values for which we know f(x), because, according to the information here, we have values of f(x) when x is an even integer, while we don't know the behavior of f(x) when x is an odd integer, and also we don't know the behavior for real numbers in general.. hope that was helpful.

And what about the question which says
C. For what values of x does f'(x) appear to be negative?

 

[beastforever]

And what about the question which says
C. For what values of x does f'(x) appear to be negative?

that very much depends on the behavior of the function, one way is to try and get the equation of the function itself, which i haven't been able to do yet..., is it given in the question that the function is defined only for even values of x?

 

[GaussianSurface]

that very much depends on the behavior of the function, one way is to try and get the equation of the function itself, which i haven't been able to do yet..., is it given in the question that the function is defined only for even values of x?

Yes, it is. Actually it says Suppose that f is given for x in the interval [0,12]

 

[Svein]

Let me see if I got you said:
then basing on what you said I'd got dividing the differences
=> -3/2 first value for f
=> -1/4 second value for f
=>-1/6 third ''
=> 2/8 = 1/4fourth
=> 3/10 fifth
=> 3/12 = 1/3 sixth
what you said about the infinity I didn't understand what you meant I'm kind of lost on those last two problems, help please.

First: Check your arithmetic. You have errors in the subtractions! The divided differences are:
-1.5
-0.5
0.5
1
1.5
1.5

Second: Best estimate for f'(2) is actually (f(4) - f(0))/4 = -1

Third: Assuming that the secants (divided differences) are reasonable approximations to the derivatives, take a look at them and then answer questions B and C.