5. find the slope intercept equation of the line · 5. find the slope intercept equation of the...

Rewrite using rational exponents.

5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).

6. Use the limit definition to find the derivative of:

7. The derivative of a function is the same thing as _____________. (HINT: see Day 1 Notes)

21. 2. 5x

x

( ) 3g x x

3

34

53. 8 4 4.x

x

Rewrite using rational exponents.

5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).

6. Use the limit definition to find the derivative of:

7. The derivative of a function is the same thing as the SLOPE of a function at a point.

21. 2. 5x

x

( ) 3g x x

3

34

53. 8 4 4.x

x

3

2 3x

12x1

2( 5)x 1

3(8 4 )x3

45x

int : 5 3( 4)po slope y x int : 3 7Slope y x

'( ) 0f x

3( )g x x2'( ) 3g x x

4( ) 2h x x3'( ) 8h x x

5 3( ) 3 2f x x x 4 2'( ) 15 6f x x x

( ) 7f x

If n is a rational number, then the function

f(x) = xn is differentiable and

1n ndx nx

dx

Differentiable

means you can take the derivative!

3: ( ) 5Ex f x x

2( ) 15d

f x xdx

The derivative of a constant function is 0. That is, if c is a real number, then

What type of lines do constants make?

0d

cdx

'( ) 0

: ( ) 4E f

f

x x

x

Constants are horizontal lines – and their slope is zero. Remember, derivatives come

from slope.

Find each derivative using the power rule.

21. ( ) 4f x x 62. ( ) 3 7f x x

2

23. ( )f x

x 3

74. ( )f x

x

5. ( ) 3g x x 3 46. ( ) 2f x x

Write answers with only positivewhole exponents and radicals!

Find each derivative using the power rule.

21. ( ) 4f x x 62. ( ) 3 7f x x

2

23. ( )f x

x

3

74. ( )f x

x

5. ( ) 3g x x 3 46. ( ) 2f x x

4

21

x

3

2 x

1

38

3

x

8x 518x

3

4

x

Write answers with only positivewhole exponents and radicals!

The sum (and difference) of two differentiable functions is differentiable and is the sum (or difference) of their derivatives.

( ) ( ) '( ) '( )d

f x g x f x g xdx

( ) ( ) '( ) '( )d

f x g x f x g xdx

38. ( ) 4 5f x x x

439. ( ) 3 2

2

xf x x x

4 37. ( ) 3 2f x x x

4 310. ( ) 2 4f x x x x

Write answers with only positivewhole exponents and radicals!

38. ( ) 4 5f x x x

439. ( ) 3 2

2

xf x x x

4 37. ( ) 3 2f x x x

4 310. ( ) 2 4f x x x x

3 212 6x x 23 4x

3 22 9 2x x 3 2 2

4 6x xx

Write answers with only positivewhole exponents and radicals!

The equation for the slope of the line tangent to the curve.

Is the slope of the line the same as we go across a curve?

We can substitute in different x-values for our derivative equation to find the slope at specific points.

Write this down!

Question 1:

Slope of a graph at a specific point, c. Find the derivative (difference quotient)

then substitute in c for x and simplify.

Question 2:

Finding a formula for the slope at anypoint on the graph Find the derivative

Question 3:

Finding the equation of the tangent line for at a specific point on the graph Use

1 1( )dy

y y x xdx

Find the slope of the graph of f(x)=x4 when:

a. x=-1 b. x=0 c. x=1

f’(x)=

a. f’(-1)=

Remember a derivative

is slope!

f’(x)=

b. f’(0)=

f’(x)=

c. f’(1)=

Find the slope of the graph of f(x)=x4 when:

a. x=-1 b. x=0 c. x=1

f’(x)=

a. f’(-1)=

Remember a derivative

is slope!

f’(x)=

b. f’(0)=

f’(x)=

c. f’(1)=

34x 34x34x

4 0 4

Given f(x) = 3x2 + 5x. Write the equation of the tangent line at x = 2.

First, find f’(x).

Then, find f’(2).

In other words, the derivative of f(x) is _______. The slope of the tangent line at x = 2 is ____.

Can you find the equation of the tangent line?? *Substitute the x-value into the original equation to find y!!You need (x1, y1) or (x1, f(x1))

Given f(x) = 3x2 + 5x. Write the equation of the tangent line at x = 2.

We could say f’(x) = 6x + 5.

So we could say f’(2) = 17.

In other words, the derivative of f(x) is 6x + 5. The slope of the tangent line at x = 2 is 17.

Can you find the equation of the tangent line??

22 17( 2)

17 12

y x

y x

*Substitute the x-value into the original equation to find y!!You need (x1, y1) or (x1, f(x1))

1. Find the coordinate point. (x1, y1)Substitute the given x-value into the ORIGINAL function to find the y-value of the point

2. Find the slope of the line. m = slopeTake the derivative of the function. Then substitute the given x-value into the derivative to find the slope at that point.

3. Use point-slope formula with the slope and point that you found! y – y1 = m(x – x1) where your point is (x1, y1) and your m = slope

First, find the coordinate pair.

Next, find the derivative of the function.f(x) = -2x2 + 9x + 1

Next, find the slope of the tangent line at x = 3.

Finally, find the equation of the tangent line at the point (3, ___).

First, find the derivative of the function.

f(x) = -2x2 + 9x + 1

Next, find the slope of the tangent line at x = 3.

Finally, find the equation of the tangent line at the point (3, 10).

'( ) 4 9f x x

3m

10 3( 3)

3 19

y x

y x

'(3) 4(3) 9f

Ex. Find an equation of the tangent line to the graph of f(x) = x2 when x = -2.

Remember to write an equation of a line we need slope and a point!

Ex. Find an equation of the tangent line to the graph of f(x) = x2 when x = -2.

f’(x)=

Remember to write an equation of a line we need slope and a point!

4m

2x *Substitute the x value into the original equation to find y!!

Write this down!

4 4( 2)y x

( 2) 4f

Have a great day!

Not used Fall ‘18