## anonymous one year ago ....

1. anonymous

Does that say $$\large f(x) = 4x + e^2x$$

2. anonymous

4x+e^2x

3. anonymous

e^(2x)

4. anonymous

Yes

5. mathmate

@camila1315 or is it: $$\large f(x) = 4x + e^{2x}$$

6. anonymous

Yes thats it

7. anonymous

The local linear approximation at x = a is essentially the tangent line approximation at the point. If x ≈ a , then f(x) ≈ f(a) + f '(a) (x-a)

8. anonymous

The actual error $$\Delta y$$ can be approximated as follows $$\large \Delta y \approx \Delta x \left[ \frac{dy}{dx} \right]_{x=a}$$

9. anonymous

delta x here is 1.1 - 1 = .1 dy/dx = 4 + e^(2x) * 2 plug in x = 1 estimated delta y = .1 ( 4 + 2*e^(2*1))=1.8778 actual delta y is f(1.1) - f(1) = 2.035957 the percent error of this approximation is ( 2.035957 - 1.8778) / (2.035957) * 100 = 7.77%

10. mathmate

I suggest to compare the approximate and accurate values of f(1.1). With $$\large f(x) = 4x + e^{2x}$$ $$\large f'(x)=2e^{2x}+4$$ f(1.0)=11.3891 exact=f(1.1)=13.4250 approx=f(1.0)+(1.1-1.0)*f'(1.0)=13.2669 approx/exact=13.2669/13.4250=0.9882 i.e. the approximate is short by 1.18% |dw:1440283809059:dw| when we examine the curve of f(x) above, it is obvious that the linear approximate understimates the value of f(1.1) because the slope is constantly increasing.

11. mathmate

*linear approximation

12. anonymous

thats odd we get two different answers your reasoning looks valid, but so does mine. i will be back later and try to figure out the discrepancy

13. anonymous

I have answer choices if you have different answers A. Between 0% and 4% B. Between 5% and 10% C. Between 11% and 15% D. Greater than 15%

14. anonymous

my result came to be between 5% and 10% I can show you my reasoning again

15. mathmate

I don't think it is a difference of the math, it's a difference in the interpretation of the question: " the % error of $$this$$ approximation is" I interpret it as asking for the error of f(1.1), and you interpret it as asking for the error of f(1.1)-f(1).

16. anonymous

$\large \Delta y \approx \Delta x \left[ \frac{dy}{dx} \right]_{x=a} \\ ~ \\ \large \sf \% error = \left |\frac{\Delta y - \Delta x \left[ \frac{dy}{dx} \right]_{x=a} }{\Delta y }\right| \cdot 100$

17. anonymous

Yeah I think math is right

18. anonymous

let me know if you get it correct. I don't think its right.

19. anonymous

Okay I will. Thank you both for all the help!!!