Page 81 - Calculus Workbook For Dummies

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Chapter 5: Getting the Big Picture: Differentiation Basics

f Draw a function containing three points where — for three different reasons — you would not
be able to determine the slope and thus where you would not be able to find a derivative.
Your sketch should contain (1) Any type of gap or discontinuity. There’s no slope and thus no
derivative at a gap because you can’t draw a tangent line at a gap (try it). (2) A sharp point or
cusp. It’s impossible to draw a tangent line at a cusp because a line touching the function at
such a sharp point could rock back and forth. So there’s no slope and no derivative at a cusp.
(3) A vertical inflection point. Although you can draw a tangent line at a vertical inflection
point, since it’s vertical, its slope — and therefore its derivative — is undefined.
g Use the difference quotient to determine the derivative of the line y = 4 x - 3. y’ = 4.

4 ^ x + h - - ^h 3 4 x - 3h
y = l lim
h " 0 h
3
4 x + 4 h - - 4 x + 3
= lim
h " 0 h
4 h
= lim
h " 0 h
= lim4
h " 0
y = l 4
You can also figure this out because the slope of y = 4 x - 3 is 4.
2
h Use the difference quotient to find the derivative of the parabola f x = 3 x . f'(x) = 6x
^ h
2 2
h -
3^ x + h 3 x
x =
f l ^ h lim
h " 0 h
2
2
3 _ x + 2 xh + h i - 3 x 2
= lim
h " 0 h
2
2
3 x + 6 xh + 3 h - 3 x 2
= lim
h " 0 h
6 xh + 3 h 2
= lim _ Now, factor out the hi
h " 0 h
h 6 + 3 hh
x
^
= lim ^ Cancel the hh
h " 0 h
x
= lim 6 + 3 hh _ Now plug in 0i
^
h " 0
= 6 x + $
3 0
x =
f l ^ h 6 x
2
i Use the difference quotient to find the derivative of the parabola from problem 4, y = - x + 5.
y' = –2x.
2 2
h + - -
-^ x + h 5 _ x + 5i
y = l lim
h " 0 h
2
2
2
-_ x + 2 xh + h i + 5 + x - 5
= lim
h " 0 h
2
2
2
- x - 2 xh - h + + x - 5
5
= lim
h " 0 h
- 2 xh - h 2
= lim ^ Now factorh
h " 0 h
h - 2 x - hh
^
= lim ^ And cancelh
h " 0 h
= lim - 2 x - hh
^
h " 0
l
y = - 2 x