Finding Unknown Constants for a Cubic Function with Given Derivatives

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In summary, the student attempted to find the constants a,b,c, and d so that the function f(x)=ax^3+bx^2+cx+d has its first derivative equal to 4 at the point (1,0) and its second derivative equal to 5 at the point (2,4). He found the first and second derivative but was unable to solve for the coefficients because the two equations were not related.
  • #1
MrJamesta
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Homework Statement


determine the constants a,b,c, and d so that the function f(x)=ax^3+bx^2+cx+d has its first derivative equal to 4 at the point (1,0) and its second derivative equal to 5 at the point (2,4)

Homework Equations

The Attempt at a Solution


I found the first and second derivative
f'(x)=3ax^2+2bx+c
f''(x)=6ax+2b

I set
5=12a+2b
I also set
4=3a+2b+c
I get stuck trying to find the two different variables when working with the second derivative first. I know I have to substitute for the unknowns, but I don't know where to start.
 
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  • #2
Tricky problem but if a function f(x) has a derivative at some specific point then what can you say about f(x)?
 
  • #3
That f(x) is continuous at that specific point?
 
  • #4
Yes, that's true, but I am thinking even more obviously than that?
 
  • #5
  • #6
You tell me.
 
  • #7
Oh, it's an x intercept.
 
  • #8
No, it's not.

f(1)=?

You have 4 unknowns so you need 4 equations to solve this. You came up with two of them. What are the other two? It's pretty obvious.
 
  • #9
SammyS said:
What is f(1) ?
MrJamesta said:
Oh, it's an x intercept.

paisiello2 said:
No, it's not.
Just to be clear, and to reduce confusion on the part of the OP, 1 is an x-intercept.
 
  • #10
Well, what is an x-intercept? I think it is the value of the function when x=0. But f(1) is the value of the function when x=1. So it is not an x-intercept.

Regardless, it is irrelevant to solve the problem. What are the other two equations?
 
  • #11
paisiello2 said:
Well, what is an x-intercept? I think it is the value of the function when x=0.
No, that's the y-intercept, a point on the y-axis.
paisiello2 said:
But f(1) is the value of the function when x=1. So it is not an x-intercept.

Regardless, it is irrelevant to solve the problem.
But your reply to the OP was incorrect, possibly steering him/her in the wrong direction.
 
  • #12
To get back to the original question:
MrJamesta said:
Oh, it's an x intercept.
Right. Can you use it to find another equation?
 
  • #13
I found
0=a+b+c+d
4=8a+4b+2c+d
 
  • #14
MrJamesta said:
I found
0=a+b+c+d
4=8a+4b+2c+d
Now, use the other two equations and solve for the unknown coefficients.
 

Related to Finding Unknown Constants for a Cubic Function with Given Derivatives

1. How do you find unknown constants?

There are several methods for finding unknown constants, depending on the type of equation and the type of constant. One common method is to use experimental data and mathematical models to calculate the value of the constant. Another method is to use known relationships between different constants to solve for the unknown one.

2. What are some common techniques for determining unknown constants?

Some common techniques for determining unknown constants include regression analysis, numerical methods such as curve fitting, and trial and error methods. These techniques can be used to analyze experimental data and derive relationships between variables and constants.

3. How do you ensure the accuracy of the determined constant?

In order to ensure the accuracy of a determined constant, it is important to use reliable and precise data, as well as robust mathematical models. It is also important to validate the results using multiple methods and to consider potential sources of error.

4. Can known constants change over time?

In general, known constants do not change over time. However, there are some constants that may vary slightly depending on factors such as temperature or pressure. In these cases, it is important to account for these variations in order to accurately determine the value of the constant.

5. How do unknown constants affect the validity of scientific theories?

Unknown constants can have a significant impact on the validity of scientific theories, as they can change the predicted outcomes of experiments and observations. It is important for scientists to accurately determine these constants in order to refine and improve existing theories, or to develop new ones.

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