Math 122 Exam 2 Review Session

Derivative Rules

100

Find the First derivative of y=9√x

^{Explain what rules you used to get the answer}

9/2 x^-1/2

200

find the first derivative of 

t= z³+ 4/z² + 6z^-3

^{Explain what rules you used to get the answer} 

3z² - 8z^-3 - 18z^-4

300

Explain the chain rule in your own words 

plug the inside function into the derivative of the outside function

Multiply that by the derivative of the inside funciton 

400

W(s) = 6e^√s

3e^√s (s^-1/2)

500

Find the derivative of f(t) = 5t e^5-2t

5e^5-2t - 10e^5-2t

Local Max and Minimum

100

What is a critical point?

What is an inflection point? 

critical point - when the first derivative is 0 

Inflection point - when the second derivative is 0 

200

Explain the first and second derivative tests. Which should you use first and why?

1st derivative test: if the first derivative changes from negative to positive, then F has a local minimum. if the first derivative changes from positive to negative, then f has a local maximum. 

2nd derivative: plug the critical points into the second derivative and observe the behavior. 

second derivative first because its quicker. if you get 0 as an answer, it tells you nothing about the function and you then have to try the first derivative test.

300

Find all of the critical points of f(x)= 2x³ + 9x² + 12x. Identify critical points as local maximum, minimum, or neither. Find any inflection points. 

-2 = local max 

-1 = local min

-1.5 = inflection point 

400

f(x) = 2x³ + 3x² - 36x + 5

Find all critical points and inflection points. classify each as a local max, min, or neither. 

CP = -3, 2

IP = -12 

-3 = local max 

2 = local min

500

Find the critical points and identify the local max and min for f(x)=x³-9x²-48x+52 

CP = -2, local max 

CP = 8, local min

Global max and Minimum

100

Identify the global and local max and mins in the graph I will draw on the board. 

TBD :)

200

Identify the steps in finding global max and min

1. identify if it is on a closed or open interval 

2. find the first derivative 

3. evaluate at the endpoints (if present) and critical points 

4. if open interval,  classify critical points and sketch 

5. determine max and min 

300

f(x) = x³ - 3x² - 9x +15 

determine global and local max and mins

-1 = global max 

3 = local min 

-5 = global min 

-140 = global min 

400

find the exact global max and min 

g(x) = 4x-x²-5 

x= 2, global max 

500

Find the exact global max and min

g(t) = te^-t for t>0

No solution 

Profit, cost, and revenue

100

Marginal revenue is really finding what? 

The first derivative / slope

200

a manufactoring company has a cost function c(q)=3000+20q and a revenue function of

R(q) = - q²+1000q

find the quantity that maximized profit 

what is that profit? 

q = 490 

Profit = 237,100

300

TRIVIA !!!
Forward, I am heavy; backward, I am not. What am I?

TON 

400

Revenue is given by r(q)=450g and cost is given by c(q)=10,000+3q² at what quantity is profit maximized? what is the total profit at this production? 

Max profit occurs at q=75 

the total profit will be $6875

500

A water park finds that with an admission price of $12, attendance is 400 per day. For every $1 decrease in price, 40 more people visit the park per day. What is the park attendance when admission prices are set to maximize revenue (to the nearest person)? 

$11 

More derivatives :)

100

What is the derivative of a^x

What is the derivative of e^x

What is the derivative of ln(x) 

ln(a) x a^x

e^x

1/x

200

given f(x) = e^-x + x , find the derivative 

-e^-x + 1

300

Find the derivative of y= t² ln(t)

2tln(t) + t²/t

400

find the equation of the tangent line 

y= (2x+4)² at x=1 

y=-8x + 12

500

find the equation the line tangent to f(x)=7xlnx at the point where x=1 

y=7x-7