[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
= 8x multiply and add
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
_____
= 1/(2√x + 3) add 0 and add like terms
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