Calculus: The Language of Change

Exercise Set 3.2
Graphical Increments

by  Michelle Nguyen

Exercise 2 : Measuring the Derivative

Views of a function y= f[x] are  show in the three  figures below by 50 and focused on the  graph above three  different  x's.  Measure the  derivative  at each point by finding the slope of the  approximating  straight  line
 (page 47).

Solution  to  the  first  part

y= f[x] magnified at x=1, f'[1]=?

Take  two approximate  points on the tangent  line  at  x=1.
Consider these two (dx, dy) points:  (0, 0) and ( 0.01,0.02).

f'[x]=  ((((Subscript[y, 2]) - (Subscript[y, 1])))/(((Subscript[x, 2]) - (Subscript[x, 1]))))

f'[1]  = (((0.02 - 0))/((0.01 - 0)))\(TextForm\`	\(f'\)[1] = 0.02\/0.01\)So,  f'[1]  =   2

Solution   to  the  second  part

f'[-(1/2)]=?

Take  two approximate  points on the tangent  line  at  x= -1/2.
Consider these two (dx, dy) points: (0, 0), (-0.02,  0.02). 

f'[x]=  ((((Subscript[y, 2]) - (Subscript[y, 1])))/(((Subscript[x, 2]) - (Subscript[x, 1])))) 

f'[-(1/2)] =(0.02/(-0.02))

f'[-(1/2)]  =  -1

Solution  to the third part

f'[(3/2)]=?

Take  two approximate  points on the tangent  line  at  x= 3/2.
Consider these two (dx, dy) points: (0, 0),  (0.00667, 0.02). 

f'[x]=  ((((Subscript[y, 2]) - (Subscript[y, 1])))/(((Subscript[x, 2]) - (Subscript[x, 1]))))

\(TextForm\`\(\(f'\)[3\/2]\  = 0.02\/0.00667
\)\)

f'[(3/2)]  =  3