onegirl 2 years ago Write the equation (estimated) of the tangent line.h(x) = f(x)g(x) at x = 1.

1. onegirl

@rizwan_uet can u help?

2. Mertsj

Would you please post the problem correctly?

3. onegirl

okay

4. onegirl

\[h(x) = f(x)g(x) \] at x = 1

5. onegirl

@tcarroll010 can u help?

6. onegirl

7. Mertsj

How is anyone supposed to help you when we have no idea what the functions are?

8. onegirl

well thats the question, it says find an equation of a tangent line to h(x) = f(x)g(x) at x = 1

9. onegirl

there are no functions..

10. onegirl

so can anyone help?

11. tcarroll010

You can get the first derivative of h(x) as follows with the product rule: h'(x) = f'(x)g(x) + f(x)g'(x) That will be your slope at any point where the first derivative is defined. Now, you use the point-slope formula for the equation of a line:\[y - y _{1} = m(x - x _{1})\]and here, "m" is as defined above with the first derivative for your slope. Make sure you substitute "1" where you see "x" in that first equation. x1 is 1 and y1 is h(1). So, just make the substitutions and that is the equation for the tangent line.

12. onegirl

I did this but my teacher told me to write the equation an estimated one "the curve is (1, h(1)) , using the product rule h'(x) = f'(x) g'(x) = + g(x)f'(x) substituting x= 1 into the 1st derivative the slope will be h'(1) = f(1) g'(1) + g(1)f'(1) so the slope h'(1) and the point (1,h(1) so y = (f'(1)g(1) +f(1)g'(1) (x - 1) + f(1)g(1)

13. onegirl

so first i have to get the derivative of h'(x) f'(x) g(x) + f(x)g'(x) right?

14. tcarroll010

his is just what I said also. Exactly what I said.

15. onegirl

okay

16. tcarroll010

The only difference, which is really nothing, is that I said y1 is h(1). That is the same as f(1)g(1). Just written slightly differently, that's all. And that's all you have to do. And you put the h(1) on the right which is fine. Same answer.

17. onegirl

Well i did this but my teacher said i need to write an equation and estimated one. :/

18. onegirl

so maybe my final answer needs to be estimated?

19. tcarroll010

Yours and my answer, which is the same and is the correct and exact answer. Let me see if we can estimate this for x = 1. Hold on a minute.

20. onegirl

okay

21. tcarroll010

This can be re-written: y = [f'(1)g(1) + f(1)g'(1)](x - 1) + f(1)g(1) y = [f'(1)g(1) + f(1)g'(1)]x + ( f(1)g(1) - [f'(1)g(1) + f(1)g'(1)] ) Now, if the derivative at x = 1 is small, we could drop it out of the far right side of the above equation as it is in the form: y = mx + b y = [f'(1)g(1) + f(1)g'(1)]x + f(1)g(1) But that is if the derivative at x = 1 is small. Then, we have a good approximation.

22. onegirl

okay

23. onegirl

well i'll try this thanks

24. tcarroll010

If I were the teacher, this is what I'd be looking for. Remember the stipulation of the derivative being small.

25. onegirl

okay

26. tcarroll010

Good luck to you in all of your studies and thx for the recognition! @onegirl And you're welcome!