Help me find the derivative algebraically

Question

The problem is g(t)= t²+t at t=-1

So I approached it like I normally do.

1)   g'(-1)=lim_h→0 (g(-1+h)-g(-1))/h

2) g'(-1)=lim_h→0 (g(-1+h)²+(-1+h) -g(-1))/h

3) g'(-1)=lim_h→0 (1-2h+h²-(-1))/h

But once I simplify by substituting 0 for h I get 1, and my professors answer is -1.

Please help me figure out what I did wrong!

Answer 1

Hi Olivia,

 

Your goal is to get rid of that h in the denominator because we cant have a zeron in the denominator, so you must do some algebra first.

 

= g(t+h) - g(t)

   ---------------

            h

= (t + h) ² + (t+h) - (t² + t)

   -------------------------------

                     h

=  t² + 2th + h² + t + h - t² - t

    ---------------------------------

                         h

combine like terms, you will have

 

= h² + 2th + h

   ---------------

            h

factor out h

 

= h ( h + 2t + 1)

   -----------------

              h

the h on top and bottom cancel out, you are left with,

 

= h + 2t + 1

 

no more h in the denominator, we are good to substitute, evaluate at h=0 and t= (-1)

 

= 0 + 2(-1) + 1

 

= 0 - 2 + 1

 

= -1

 

Hope it helps.

 

Answer 2

            lim    (t+h)²+(t+h) -t²-t     

 g'(t) = h→0  ----------------------
                              h

 

 

            lim      t²+2th+h²+t+h-t²-t      lim        2th + h² + h

 g'(t) =  h→0  -------------------------- =   h→0   ----------------------  

                                  h                                            h                          

 

 g'(t) =  lim   (2t + h + 1)  =  2t +1

            h→0

 

g'(-1) = 2(-1) + 1 = -2 + 1 = -1

Answer 3

Lets continue to step 2.

 

lim        (-1 + h)² + (-1 + h) - [(-1)² - 1]

h->0 ______________________________    =

                                  h

 

 

Expand the numerator and getting rid of parentheses and brackets.

 

 

lim     (1 - 2h + h²) - 1 + h - 1 + 1

h->0  ________________________      

                               h

 

 

The ones cancel each other out.

 

lim      -2h + h² + h

h->0   ____________

                    h

 

 

Now we can factor out an h in the numerator.

 

lim      h(-2 + h + 1)

h->0  ___________

                   h

 

 

Cancel out h.

 

 

lim     -2 + h + 1

h->0

 

 

Plug in h=0 into the limit as the final step.

 

-2  + 1 = -1