## anonymous one year ago For f(x)=-2x^2 + 3x + 1 find (f(2+h)-f(2))/h Simplify completely and check your solution.

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1. anonymous

f(2) and upon solving its answer is -1 so basically the end product is -2h-5

2. anonymous

but here's the catch I don't know how to check this thing ?

3. Nnesha

f(2+h) means x = 2+h so you need to find f(2+h) replace x by 2+h in $$\color{red}{\rm f(x)}$$ function $\large\rm f(2+h) = -2\color{ReD}{(2+h)^2} +3\color{reD}{(2+h)}+1$

4. SolomonZelman

$$\large\color{black}{ \displaystyle \frac{ \color{red}{f(h+2)}-\color{blue}{f(h)} }{h} }$$ $$\large\color{black}{ \displaystyle \frac{ \color{red}{-2(h+2)^2+3(h+2)+1}-\color{blue}{-2h^2 + 3h + 1} }{h} }$$

5. SolomonZelman

Correction: $$\large\color{black}{ \displaystyle \frac{ \color{red}{-2(h+2)^2+3(h+2)+1}-\color{blue}({-2h^2 + 3h + 1)} }{h} }$$

6. anonymous

I repeat it's f(h+2)-f(2)/h not f(h+2)-f(h)/h

7. Nnesha

it should be like this $\large\rm \frac{( -2\color{Red}{(2+h)^2}+3\color{Red}{(2+h)}+1) - (2\color{ReD}{(2)}^2+3\color{ReD}{(2)}+1)}{ h }$

8. anonymous

break it into pieces: the difference quotient $$D(af+bg)=aD(f)+bD(g)$$ so let's look at how it beahves on $$x^2,x,1$$ independently: $$D(x^2)=\frac{(h+2)^2-2^2}h=\frac{h^2+2h\color{red}{+4-4}}h=\frac{h^2+2h}h=h+2\\D(x)=\frac{(h+2)-2}h=\frac{h\color{red}{+2-2}}h=\frac{h}h=1\\D(1)=\frac{1-1}h=\frac0h=0$$

9. anonymous

yes correct nnesha but the thing is upon solving I got the final answer as (-2h-5) but could someone tell me how to check this thing ? It's given in the chapter of functions.

10. Nnesha

$$\color{blue}{\text{Originally Posted by}}$$ @Nnesha it should be like this $\large\rm \frac{( -2\color{Red}{(2+h)^2}+3\color{Red}{(2+h)}+1) - (2\color{ReD}{(2)}^2+3\color{ReD}{(2)}+1)}{ h }$ $$\color{blue}{\text{End of Quote}}$$ correction it should be like this $\large\rm \frac{( -2\color{Red}{(2+h)^2}+3\color{Red}{(2+h)}+1) - (-2\color{ReD}{(2)}^2+3\color{ReD}{(2)}+1)}{ h }$

11. anonymous

oops, that should read $$h^2+4h$$ on top which gives $$D(x^2)=h+4$$

12. anonymous

I appreciate the help Oldrin but we have not covered the "difference quotient" topic yet but again as aforementioned this question is from the chapter of functions.

13. anonymous

now put it all together: so then it follows that $$\frac{f(h+2)-f(2)}h=D(-2x^2+3x+1)\\\qquad\qquad\qquad\quad=-2\cdot D(x^2)+3\cdot D(x)+D(1)\\\qquad\qquad\qquad\quad=-2(h+4)+3(1)+0\\\qquad\qquad\qquad\quad=-2h-8+3\qquad\qquad\qquad\qquad=\color{green}{-2h-5}$$

14. anonymous

Correct but how to check this thing ? I'm done trying after one hour so that's why came here today.

15. anonymous

to check, you could plug in values of $$h$$, say, $$h=1$$: $$\frac{f(1+2)-f(2)}1=f(3)-f(2)=\ ?=-2(1)-5=-7$$and then check to make sure $$f(3)-f(2)=-7$$

16. anonymous

and then similarly with $$h=2$$, maybe $$h=3$$, etc.

17. anonymous

okay and then what exactly does h=2 or h=3 check ? I mean the end result would be different as well.

18. anonymous

well, you would check that: $$\frac{f(2+2)-f(2)}2=\frac{f(4)-f(2)}2=\,? =-2(2)-5=-9$$ so you'd verify that $$\frac12(f(4)-f(2))=-9$$ for $$h=2$$

19. anonymous

Oldrin, the answer which you gave is entirely different than the main question. The tricky part with checking the solution is you have to tie it back to the main equation. What you told is giving a whole new answer entirely.

20. Nnesha

well i will check my work again .... hm

21. Nnesha
22. anonymous

Net could you pls explain me that in the form of equations here, the thing is I'm not used to paragraph cum equation type of explanations.

23. anonymous

*nnesha

24. anonymous

@abs202 no, you are very confused, unfortunately, but i don't think i'm capable of helping correct that right now

25. UsukiDoll

For f(x)=-2x^2 + 3x + 1 find (f(2+h)-f(2))/h $f(x) = -2x^2+3x+1$ we have to use the definition of the derivative which is $\large \frac{f(x+h)-f(x)}{h}$ so our x = 2 for $\large \frac{f(2+h)-f(2)}{h}$ If we just have x = 2 in the function we will just have $f(2) = -2(2)^2+3(2)+1 =-2(4)+6+1=-8+6+1=-2+1=-1$ so that portion is correct now we use the definition of the derivative to solve $f(x) = -2x^2+3x+1$ $\large \frac{-2(2+h)^2+3(2+h)+1-(-1)}{h}$ $\large \frac{-2(h^2+4h+4)+6+3h+1+1)}{h}$ $\large \frac{-2h^2-8h-8+6+3h+1+1}{h}$ rearrange the terms $\large \frac{-2h^2-8h+3h-8+6+1+1}{h}$ $\large \frac{-2h^2-8h+3h}{h}$ $\large \frac{-2h^2-5h}{h}$ Now we can factor an h and cancel it. $\large \frac{(h)[-2h-5]}{h}$ so our final answer is $\large -2h-5$

26. UsukiDoll

The only thing I can think of to make sure that you have -2h-5 is by doing the process all over again and if you have the same answer, then the arithmetic was done correctly. If there are different answers, then there was an error.

27. UsukiDoll

There is another method though. In Calculus I, the definition of the derivative is taught first before all the fancy shortcut derivatives come into play. Using the definition of the derivative and those derivative techniques should yield the same answer, but the problem is that the derivative of $f(x) = -2x^2+3x+1$ is $f'(x) = -4x+3$ ??????????????

28. UsukiDoll

I'll provide an example sort of like a proof that derivative techniques and the definition of the derivative yield the same result. Consider the function $\large f(x)=5x^2+3x-7$ The derivative is $\large f'(x)=10x+3$ now compare this to the attachment

29. UsukiDoll

Consider the function $\large f(x)=5x^2+3x-7$ since -7 is a constant, the derivative is 0 The derivative is $\large f(x)=5(2)x^{2-1}+(1)3x^{1-1}+)$ $\large f(x)=10x^{1}+3x^{0}$ $\large x^0 = 1$ (by zero exponent rule) $\large f'(x)=10x+3$

30. ali2x2

^ correct