Exam 3 Flashcards

1
Q

Linear Approximation Formula

A

L(x)=f’(a)(x-a)+f(a)

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2
Q

Differential f(x+∆x)

A

~f(x)+f’(x)∆x

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3
Q

Demand (price)

A

P(x)

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4
Q

Revenue R(x)

A

=xp(x)

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5
Q

Cost

A

c(x)

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6
Q

Profit

A

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

or xp(x)-C(x)

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7
Q

Critical Numbers

A

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

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

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8
Q

Increasing/Decreasing Function

A

f’(x)>0 increasing

f’(x)<0 decreasing

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9
Q

Local Max

A

increases then decreases

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10
Q

Local Min

A

decreases then increases

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11
Q

First Derivative Test

A

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.

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12
Q

Absolute Extrema

A

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).

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13
Q

Mean Value Theorem

A

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)

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14
Q

Rolle’s Theorem

A

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.

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15
Q

Second derivative and concavity

A

Convae up is when f’‘(x)>0

concave down is when f’‘(x)<0

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16
Q

Inflection point

A

Where concavity changes

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

17
Q

Second derivative test

A

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.

18
Q

Domain of an even root

A

inside>or equal to 0

19
Q

Domain with a denominator

A

can’t equal zero

20
Q

logs/ln domain

21
Q

X intercept

22
Q

y intercept

23
Q

Vertical asymptotes

A

set denominator=0 and solve after you simplify.

x=a is the form.

24
Q

Horizontal asymptotes

A

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

25
Cusp Vs. VTL
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.
26
multiplicity
even powers- same sign; odd powers change sign. If raised to a fraction, use the upper number. E.g. 4/3 is even.
27
LHopital's Rule indeterminate forms
0/0 or infinity/infinity
28
If it's a different indeterminate form for lhopital
rewrite the function.
29
lhopitals with a function raised to a function
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.
30
Optimization
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.