Calculus:  The Language of  Change

Exercise  Set  1.4
Derivatives  for  Explicit  Formulas

By Terry Sargent

Exercise  1

For the function f[x]  =  (x^3),   verify that  when x = 1.0  and  Δx = 0.5  the expression

Δ(y/Δ)x  =  
        ((f[x  +  Δx]  - f[x] )/Δ)x  =  
          4.75

but that when  x = 1.5  and Δx = 0.5  the expression

Δ(y/Δ)x  =  
        ((f[x  +  Δx]  - f[x] )/Δ)x  =  
          9.25

Solution

Let's    rewrite  the  specifics  of  our  problem  into  a  slightly  different  format  and  work from there.    Start by combining    f[x]  =  (x^3)  and   Δ(y/Δ)x  into   (((((x  +  Δx))^3)  - (x^3))/Δ)x
 then  the  algebraic  expansion of this equation  results  in:

Δ(y/Δ)x  =  
        (((x^3)  +    3  (x^2) Δx  +  
            3  x (((((Δx))^2)  )^(  )) +   
            (((Δx))^3) -  (x^3))/Δ)x

Next, we  begin  the  process  of  compacting   our  expression   step  by  step.  (This means  cancelling any terms that we can,  adding  like terms  together  or even  pulling a commom term  out from  the  equation)

Δ(y/Δ)x  =  
        ((  3  (x^2) Δx  +  
            3  x (((((Δx))^2)  )^(  )) +   
            (((Δx))^3))/Δ)x

Next,  let's   pull  out  a  Δx   from the  equation .    Δx  is   common  to  each  of  the 3  terms  in  the  numerator.   This  puts  us  in  the   position  of  now  being  able  to    cancel  out  the  Δx  contained in  the  denominator .

Δ(y/Δ)x  =  
        ((Δx 
            ( 3  (x^2)  +   
                3  x (((Δx)  )^(  )) +   
                (((Δx))^2)))/Δ)x

Now ,  that   we  have  the  right  form of  this  equation -  the amount of  work  left  in  solving this problem won't even cause us to break a sweat!

Δ(y/Δ)x  =   
      3  (x^2)  +   3  x (((Δx)  )^(  )) +   
        (((Δx))^2)

Now   set   x = 1.0     and   Δx = 0.5  and  plug  these  values  into  the  final  form  of our  equation.

3 (((1.0))^2)  +   3 (1.0) (((0.5)  )^(  )) +   
        (((0.5))^2)

so    we   get   Δ(y/Δ)x  =  4.75.

Now  if  we   set   x = 1.5   and    leave   Δx = 0.5  and   plug  these  new  values  into   our  equation:

3 (((1.5))^2)  +   3 (1.5) (((0.5)  )^(  )) +   
        (((0.5))^2)

we   now   get   Δ(y/Δ)x  =  9.25.

Exercise  2  -  computer  exercise .

Exercise  3 -  computer  exercise .

Exercise  4  -  computer  exercise .