Exam 3

Question 1

Linear Approximation Formula

Answer 1

L(x)=f’(a)(x-a)+f(a)

Question 2

Differential f(x+∆x)

Answer 2

~f(x)+f’(x)∆x

Question 3

Demand (price)

Answer 3

P(x)

Question 4

Revenue R(x)

Answer 4

=xp(x)

Question 5

Cost

Answer 5

c(x)

Question 6

Profit

Answer 6

P(x)=R(x)-C(x)

or xp(x)-C(x)

Question 7

Critical Numbers

Answer 7

In domain, when f’(x)=**0 **(Horizontal Tangent Lines)

When f’(x)=undefined (Vertical Tangent Lines or cusps)

Question 8

Increasing/Decreasing Function

Answer 8

f’(x)>0 increasing

f’(x)<0 decreasing

Question 9

Local Max

Answer 9

increases then decreases

Question 10

Local Min

Answer 10

decreases then increases

Question 11

First Derivative Test

Answer 11

Set top and bottom of derivative=0. Find critical numbers. Make number line. Find max and min based on changes in sign across interval. Remember domain!

Plug back in to find actual min/max value of function.

Question 12

Absolute Extrema

Answer 12

Find the highest (max) and lowest(min) with first derivative test critical points and using the endpoints of the interval. Plug all these x values into f(x).

Question 13

Mean Value Theorem

Answer 13

if a function is continuous on [a,b] and differentiable on (a,b), then there exists some value x=c in (a,b) such that f(b)-f(a)/b-a = f’(c).

Take f(b)-f(a)/b-a and set equal to the derivative (in derivaitve use x or c). Solve for that variable (x or c)

Question 14

Rolle’s Theorem

Answer 14

If a function is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then there exists some value x=c in (a,b) such that f’(c)=0

Take the derivative and set it equal to 0. Solve for x, that is your c value.

Question 15

Second derivative and concavity

Answer 15

Convae up is when f’‘(x)>0

concave down is when f’‘(x)<0

Question 16

Inflection point

Answer 16

Where concavity changes

Set f’‘(x)=0 and test points on the number line.

Question 17

Second derivative test

Answer 17

Take first derivative. Set f’(x)=0 for critical numbers. Plug these critical numbers into f’‘(x)

f’‘(x)>0 you have a minimum.

f’‘(x)<0 you have a maximum.

f’‘(x)=0 then can’t be determined by second derivative test.

Question 18

Domain of an even root

Answer 18

inside>or equal to 0

Question 19

Domain with a denominator

Answer 19

can’t equal zero

Question 20

logs/ln domain

Answer 20

inside> 0

Question 21

X intercept

Answer 21

set y =0

Question 22

y intercept

Answer 22

set x=0

Question 23

Vertical asymptotes

Answer 23

set denominator=0 and solve after you simplify.

x=a is the form.

Question 24

Horizontal asymptotes

Answer 24

look at the powers

powers are the same, HA is the ratio of coefficients

Bottom is bigger HA y=0

Top is bigger, no HA

Question 25

Cusp Vs. VTL

Answer 25

cusp is if the function changes signs across the first derivative at that point. VTL and not a cusp if it does not change signs across the first derivative at that point.

Question 26

multiplicity

Answer 26

even powers- same sign; odd powers change sign. If raised to a fraction, use the upper number. E.g. 4/3 is even.

Question 27

LHopital’s Rule indeterminate forms

Answer 27

0/0 or infinity/infinity

Question 28

If it’s a different indeterminate form for lhopital

Answer 28

rewrite the function.

Question 29

lhopitals with a function raised to a function

Answer 29

raise it to e^ln(function).

move e to the outside so the whole limit is raised to e. Solve the ln limit, that answer is the power on the e. Simplify.

Question 30

Optimization

Answer 30

identify the primary equation. Identify the constraint. isolate a variable in the constraint and plug it in to the primary equation. Take derivative of primary and set it equal to 0 to find the value’s max or min point. Find the other variable. Plug both back in for the acutal max or min.