Calculus Jeopardy

Unit 1

100

What is the limit as x -> 2

lim        =    -1         lim        =    5

x -> 2^-                     x -> 2⁺             

DNE

200

Find the limit as x --> 0 for the following equation:

5x² + 3x

x

5x² + 3x = x(5x + 3)

x(5x + 3)   =   5x + 3

x                  

5(0) + 3 = 3

300

Evaluate

lim           x + 3

x -> 2       x - 7

2 + 3     =    5

2 - 7            -5

= -1

400

(DOUBLE POINTS)

Evaluate: 

lim         x² - 4

x -> 2    x - 2

x² - 4 = (x - 2)(x + 2)

(x - 2)(x + 2)    =    x + 2

x - 2

2 + 2 = 4

500

Evaluate:

lim         x² - x - 6

x -> 3       x - 3

x² - x - 6 = (x - 3)(x + 2)

(x - 3)(x + 2)   =    x + 2

x - 3

3 + 2 = 5

Unit 2

100

What is the derivative of 20x⁵?

100x⁴

200

What is the derivative of 1/x²?

-2/x³

300

What is the derivative of sin(√x)?

cos(√x)/(2√x)

400

What is the derivative of 4^{x^2+3x}

ln(4)4^{x^2+3x} (2x+3)

500

(TRIPLE POINTS!!!)

What is the derivative of tan(x)^tan(x)?

tan(x)^tan(x)sec²(x)(ln(tan(x)) + 1)

Unit 3

100

   Evaluate:

lim              sin(3x)

x -> 0             x

0

0

lim         3cos(3x)  = 3cos(0)   =  3

x->0           1                1

Therefore the answer is 3

200

(DOUBLE POINTS)

Determine y^'

y= log₅(x³)

y^'= 3x²  =          3    

      In5x³           In5x

300

Find two positive numbers whose sum is 40 such that their product is a maximum. What is that maximum product? (solve using derivative)

P= x(40-x) = 40x-x²

p^' = 40 - 2x = 0  

x= 20

p= 20(40-20) = 400

Answer: 400

400

A 10 m ladder leans against a wall. The bottom is sliding away from the wall at 1 m/s. How fast is the top sliding down the wall when the bottom is 6 m from the wall?

x² + y² = 100 (x=6, y=8)

2x·(dx/dt) + 2y·(dy/dt) = 0 → 6(1) + 8(dy/dt) = 0 → dy/dt = −6/8 = −0.75 m/s

Answer: The top of the ladder is sliding down at 0.75 m/s.

500

Design the most economical open­topped barrel that will hold 20L. The cost of material per cm² for the base is triple the cost of the wall. 

(V=ㅠr²h, SA= 2ㅠr² + 2ㅠrh, 1L=1000cm³)

Unit 4

100

What is the anti-derivative of e^x?

e^x

200

What is the area under the x axis for the equation:

-\(\sqrt{49 - x^2} \)

hint(Use calc)

(pi7^2)/2

Approximately 77

300

(DOUBLE POINTS) Find the area enclosed between 1/5x +1 and -x^2 + 17

78 squared units

400

On January 1st a bacteria grows at:

B'(t)=5(2)^0.2t bacteria per day

If there were 150 bacteria on January 11th determine the population on January 21st

300/ln(2) + 150 

or approximately 583

500

ln(2)