Calc BC Units 2.1-2.6

Avg vs Inst ROC and the Defention of a derivative

100

Average rate of change of f(x)= x² 

over x =0 to x=4

f(4)-f(0) / (4-0)

16/4

4

200

Find the instant rate of change of f(x) = x² at x=1

f'(1)=2

300

find the average rate of change of f(x) = 2cos(x)

from x = 0 to x=pi/2

f(pi/2)-f(0) / (pi/2 - 0)

-1 / (pi/2) 

-2 / pi

400

Find the derivative of (x+2)²

by the definition of a derivative (come up to the board)

2(x+2)

500

Find the derivative of (2x+3)^1/2

by the definition of a derivative (come up to the board)

1/(2x+3)^1/2

Differentiability and estimating derivative at a point.

100

Is this function Differentiable at x=0?

|x|

No

200

is x² differentiable at x = 2

yes. 

It is continuous and smooth.

300

Is x^1/3 differentiable at x = 0

no, at x = 0 slope is undefined. 

400

x       4 , 6 , 8 , 10

f(x)  12, 13 , 17, 20

Find f'(5)

( f(6) - f(4) ) / (6 - 4)

(13 - 12) / (2)

1 / 2

500

t       2 , 5 , 9 , 12

v(t)  31, 26, 20, 12

v(t) models the volume of water in a barrel at time t,

Find the rate at which water enters/exits at time t=7

( f(9) - f(5) ) / (9-5)

(20 - 26) / (4)

-6 / 4

water exits the barel at 3/2 units per second

Power Rule

100

 d/dx (x²)

2x

200

d/dx (3x⁵)

15x⁴

300

d/dx (2x^1/2)

1/x^1/2

400

d/dx [(4/x) + (8/2x)]

-4/x² - 4/x²
-8/x²

500

d/dx [(8x^-1/2) + (7x^-3) + (16x⁵)]

-4/(x^3/2) - 21/x⁴ + 80x⁴

Rules, Horizontal Tangents, Differentiability of a piece wise function

100

d/dx (5)

0

200

d/dx (3x)

3

300

d/dx (x⁵ + x⁴)

5x⁴ + 4x³

400

Where does this function have a horizontal tangent line? (full coordinate)

f(x) = x²-6x

(3, -9)

500

Is this piecewise function differentiable at x = -1? 

f(x) = x² for x<-1

f(x) = x³ for x=-1

f(x) = x³ for x>-1

no 

d/dx (left) does not equal d/dx (right)

Trivia Column

100

What is the final boss in Minecraft

The Ender Dragon

200

What is the famous line Bugs Bunny says

(You must do a Bugs Bunny accent)

Eh - What's up, Doc

300

At what temperature are Celsius and Fahrenheit the same?

-40 degrees.

400

Who was the Knicks MVP in Game 4 of the NBA Finals

De-aron Fox

500

If I had a nickel for every time Tiger Woods was arrested (not convicted) for a DUI charge, how many nickels would I have?

(Hint: there are two parts to this answer)

I'd have two nickels,

Which isn't a lot, but it's strange that it's happened twice.