[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