[tangent lines][section 3]

(1.)  f(x) = -4x + 6;  (3, -6)          here is the problem

f(x + h) = -4(x + h) + 6    replace x with x + h

f(x + h) - f(x)
f'(x) = lim _______________       use this formula
h->0       h

=       -4(x + h) + 6 - (-4x + 6)  make substitutions
lim  ___________________________
h->0            h

=        -4x - 4h + 6 + 4x - 6           multiply thru
lim  ________________________
h->0           h

=   lim   -4h/h             combine like terms
h->0

=   lim  -4                   cancel
h->0

=    -4

y - y1 = m(x - x1)   use this line formula

y + 6 = -4(x - 3)    make substitutions

y + 6 = -4x + 12       multiply thru parentheses

-6        -6         subtract 6 from each side
___________________
y    = -4x + 6          subtract

result: y = -4x + 6

(2.)  g(x) = 15(x - 3) + 40  ;  (3, 40)

g(x) = 15x - 45 + 40           multiply thru parentheses

g(x) = 15x - 5                combine like terms

g(x + h) = 15(x + h) - 5     replace x with x + h
g(x + h) = 15x + 15h - 5        multiply thru

g(x + h) - g(x)
g'(x) = lim ______________________   use this formula
h->0       h

=         15x + 15h - 5 - (15x - 5)     make substitutions
lim      _________________________
h->0           h

= lim  15h/h              combine like terms
h->0
=  lim 15                   cancel
h->0

= 15

y - y1 = m(x - x1)  use this line formula

y - 40 = 15(x - 3)   make substitutions

y - 40 = 15x - 45      multiply thru parentheses

+  40       + 40       add 40 to each side
____________________
y    = 15x - 5         add

result:  y = 15x - 5

(3.)  f(x) = 2(x - 3)2;  (4,2)      here is the problem

f(x) = 2x2 - 12x + 18      multiply

f(x + h) = 2(x + h)2 - 12(x + h) + 18   replace x with x+h

f(x + h) = 2x2 + 4xh + 2h2 - 12x - 12h + 18   multiply

f(x + h) - f(x)       use this formula
f'(x) = lim ________________
h->0        h

= lim 2x2 + 4xh + 2h2 - 12x - 12h + 18 - (2x2 - 12x + 18)    make
h->0______________________________________________substitutions
h

= lim (4xh + 2h2 - 12h)/h        combine like terms on top
h->0

= lim    4x + 2h - 12           divide thru by h, cancel
h->0

=   4x + 2(0) - 12         replace h with 0

=   4x - 12                  multiply combine like terms

m = 4(4) - 12            replace x with 4

m = 4                multiply and subtracts

y - y1 = m(x - x1)  use this line formula

y - 2 = 4(x - 4)    make substitutions

y - 2 = 4x - 16      multiply thru parentheses

+  2       + 2     add 2 to each side
_____________________
y     = 4x - 14         add

result:  y = 4x - 14

(4.)  g(x) = 4x2 - 9;  (-1,-5)

g(x + h) = 4(x + h)2 - 9    replace x with x + h

g(x + h) = 4x2 + 8xh + 4h2 - 9    multiply

g(x + h) - g(x)
g'(x) = lim ________________
h->0        h           use this formula

=    lim    4x2 + 8xh + 4h2 - 9 - (4x2 - 9)  make substitutions
h->0  ________________________________
h

=    lim  (8xh + 4h2)/h       combine like terms
h->0

=   lim   8x + 4h                 divide thru by h, cancel
h->0

=   8x + 4(0)                 replace h with 0

g'(-1) = 8(-1)            replace x with -1

g'(-1) =-8              multiply

y - y1 = m(x - x1) use this line formula

y + 5 = -8(x + 1)         make substitutions

y + 5 = -8x - 8          multiply thru parentheses

-5       -5          subtract 5 from each side
___________________
y    = -8x - 13       subtract

result:  y = -8x - 13

(5.)  f(x) = -x2 + 3x + 5 ;  (1,7)

f(x + h) = -(x + h)2 + 3(x + h) + 5   replace x with x + h

f(x + h) = -x2 - 2xh - h2 + 3x + 3h + 5    multiply

f'(x) = lim  f(x + h) - f(x)    use this formula
h->0  _______________
h

= lim  -x2 - 2xh - h2 + 3x + 3h + 5 - (-x2 + 3x + 5)
h->0  ___________________________________________
h

[make substitutions]

=     lim  (-2xh - h2 + 3h)/h    combine like terms on top
h->0

=   lim  (-2x - h + 3)         divide thru by h, cancel
h->0

=    -2x - 0 + 3            replace h with 0

=   -2x + 3                 combine like terms

f'(1) = -2(1) + 3    replace x with 1

f'(1) = 1              multiply and combine like terms

m = 1              this is the slope of the tangent line

y - y1 = m(x - x1)  use this line formula

y - 7 = 1(x - 1)    make substitutions

y - 7 = x - 1           multiply thru parentheses

+  7    + 7        add 7 to each side
_______________
y     = x + 6          add

result:  y = x + 6

(6.)   g(x) = x3 ; (2,8)

g(x + h) = (x + h)3        replace x with x + h

g(x + h) = x3 + 3x2h + 3xh2 + h3   cube the binomial

g'(x) = lim g(x + h) - g(x)     use this formula
h->0 ________________
h

=  lim  x3 + 3x2h + 3xh2 + h3 - x3   make substitutions
h->0 ___________________________
h

=    lim  (3x2h + 3xh2 + h3)/h   combine like terms on top
h->0

=    lim  3x2 + 3xh + h2     divide thru by h, cancel
h->0

=   3x2 + 3x(0) + 02         replace h with 0

=     3x2                      multiply combine like terms

g'(2) = 3(2)2             replace x with 2

g'(2) = 12                multiply

m = 12           this is the slope of the tangent line

y - y1 = m(x - x1)  use this line formula

y - 8 = 12(x - 2)   make substitutions

y - 8 = 12x - 24        multiply thru parentheses

+    8        +  8      add 8 to each side
_____________________
y =    12x - 16        add

result:  y = 12x - 16

_____
(7.)  f(x) = -√x + 3; (6,-3)
_________
f(x + h) = -√x + h + 3      replace x with x + h

f'(x) = lim  f(x + h) - f(x)   use this formula
h->0  ________________
h
_________    _____
=  lim   -√x + h + 3 + √x + 3
h->0   _____________________     make substitutions
h
_____    _________
=    lim  √x + 3 - √x + h + 3           just rearrange on top
h->0  _____________________
h
_____    _________   _____    _________
=    lim   (√x + 3 - √x + h + 3)(√x + 3 + √x + h + 3)  multiply
h->0  __________________________________________top & bottom
_____    __________             by this
h(√x + 3 + √x + h + 3)

=    lim   x + 3 - (x + h + 3)  foil multiply combine like terms
h->0   _________________________  on top
_____    _________
h(√x + 3 + √x + h + 3)

=    lim        -h                        combine like terms
h->0   __________________________  on top
_____    __________
h(√x + 3 + √x + h + 3)
_____    _________
=    lim   -1/(√x + 3 + √x + h + 3)       cancel
h->0

_____    _________
=      [1/(√x + 3 + √x + 0 + 3   replace h with 0
_____

f'(6) =    1
___________        replace x with 6
______
2√6 + 3

f'(6) = 1/(2*3)             add and take sq root

f'(6) = 1/6                   multiply

m = 1/6      this is the slope of the tangent line at x = 6

y - y1 = m(x - x1)   use this line formula

y + 3 = (1/6)(x - 6)     make substitutions

y + 3 = (1/6)x - 1      multiply thru parentheses

-  3         -3   subtract 3 from each side
_______________________
y =   (1/6)x - 4         subtract

result:   y = (1/6)x - 4

_
(8.)  g(x) = 1/√x  ;  (4, 1/2)
_____
g(x + h) = 1/
x + h      replace x with x + h

g'(x) = lim g(x + h) - g(x)     use this formula
h->0________________
h
_____        _
=    lim [1/
x + h] - (1/x)      make substitutions
h->0 ___________________
h
_    _____                            _  _____
=    lim
x - x + h          multiply thru by x*x + h
h->0  _______________   and cancel as you go thru
_____  _
h[
x + h*x]
_    _____   _    _____
=    lim  [
x - x + h][x + x + h]    multiply top and bottom
h->0  _________________________________ by this
_____  _    _    _____
h[
x + h*x][x + x + h]

=     lim    x - (x + h)        foil multiply combine like terms
h->0  ___________________________
_____  _    _    _____
h[
x + h*x][x + x + h]

=    lim          -h
h->0   ______________________________    combine like terms
_____  _    _    ______
h[
x + h*x][x + x + h]

=   lim         -1
h->0   _____________________________    cancel h's
_____  _    _    _____
[
x + h*x][x + x + h]

=         -1
____________________       replace h with 0 and simplify
_
x * 2
x

=       -1
________________         replace x with 4
4 * 2
4

=    -1/16               simplify

m = -1/16     this is the slope of the tangent line at x = 4

y - y1 = m(x - x1)  use this line formula

y - (1/2) = (-1/16)(x - 4)   make substitutions

16y - 8 = -x + 4    multiply thru by 16, cancel

+ 8       +8    add 8 to each side
___________________
16y   = -x + 12    add

+x     + x           add x to each side
___________________
x + 16y = 12            add

result:  x + 16y = 12