Math Final Flashcards
You have 10 meters of fence and want to build an enclosure that is as small as possible, but it can’t be narrower than one goat (1/2 meter). How long and wide should you make the enclosure?
1) Draw & Label a Picture
2) Write down what you wish to optimize:
3) Write an equation that expresses what you wish to optimize & in terms of variables
4) Reduce equation down to 1 variable using constraint in problem
5) Differentiate & solve for Global Extrema wanted: find f(x) at critical, singular & endpoints
6) Check if realistic within domain:
w=1/2m
l=9/2
6 steps to performing Optimization
1) Draw & Label a Picture
2) Write down what you wish to optimize:
3) Write an equation that expresses what you wish to optimize & in terms of variables
4) Reduce equation down to 1 variable using constraint in problem
5) Differentiate & solve for Global Extrema wanted: find f(x) at critical, singular & endpoints
6) Check if realistic within domain
Optimization on open vs closed interval
If f (x) is continuous on a closed interval, then f(x) has a global maximum and a global minimum on that interval
A continuous function on an open interval does not necessarily have a global maximum and a global minimum on that interval. However, you can still have a local and global extrema by determining where the function is increasing and where it is decreasing.
100m of fencing to enclose a rectangular area against a straight wall. What is the largest area you can enclose?
1) Draw & Label a Picture
2) Write down what you wish to optimize:
3) Write an equation that expresses what you wish to optimize & in terms of variables
4) Reduce equation down to 1 variable using constraint in problem
5) Differentiate & solve for Global Extrema wanted: find f(x) at critical, singular & endpoints
6) Check if realistic within domain
A=1250
w=25
Extremas & How to Evaluate for Them
Found evaluating the function at critical f’(x)=0, singular f’(x)=DNE & endpoints f(a)=#, f(z)=#, interval of [a,b]
Local Max x=c if f(c)f(x) for all x near c
Local Min x=c if f(c)f(x) for all x near c
2nd Derivative Test:
- If f’‘(c)<0 then c is a local max
- If f’‘(c)>0 then c is a local min
Global Max x=c if f(c)f(x) for all x
Global Min at x=c if f(c)f(x) for all x
Find Global extrema of -x^4+x^2+1 on (-5,0)
Global Max at x=-sqrt(1/2)
Global Min at x=0
1m long string cut into 2. One piece forms a square the other into a circle. Where to cut the string to maximize the total enclosed area of both shapes?
1) Draw & Label a Picture
2) Write down what you wish to optimize:
3) Write an equation that expresses what you wish to optimize & in terms of variables
4) Reduce equation down to 1 variable using constraint in problem
5) Differentiate & solve for Global Extrema wanted: find f(x) at critical, singular & endpoints
6) Check if realistic within domain
x=1/12
Sketch x^2, x^3, logx, e^x, sinx, cosx, tanx, 1/x, sqrt(x), absolute x
See desmos
Which term in a polynomial function will dominate for small x & large x
For small x, smaller degrees of power will dominate.
For large x, larger degrees of power will dominate
Sketch & asymptotically show e^x dominates any power function
(e^x, x>k)>x^k
Polynomial Power Function
f(x)=Kx^n, K ^ n are constants, can be negative or fraction
Polynomial Functions
f(x)=anx^n+an-1x^n-1+…+a2x^2+anx+a0, a & k are constants
Sketch f(x)=x^2-x^4, f(x)=3x+x^2, e^x-x^4 using asymptotic reasoning
See desmos + notes for reasoning
Hill Function, for small & large x what does it look like
f(x)=(Ax^n)/(B+x^n)
Reason out denominator first, then simplify
For small x, f(x)=Ax^n/B
For large x, f(x)=A
Asymptotic Reasoning Process
- What is negligable at small x, simplify equation
- What is negligable at large x, simply equation
- Sketch out, find points where they intersect if required
Explain with words & graphs what lim(x>a^+/-) f(x)=L
Limit of f(x) as x approaches a is equal to L means f(x) is arbitrarily close to L provided x is sufficiently close to a (but not equal to a)
Limit of f(x) as x approaches a from below is equal to L is arbitrarily close to L provided x is sufficiently close to a and x<a (left hand limit)
Limit of f(x) as x approaches a from above is equal to L is arbitrarily close to L provided x is sufficiently close to a and x>a (right hand limit)
If function does not approach a single number as x approaches a, the limit does not exist (DNE)
Explain with words & graphs what lim(x>a^+/-) f(x)=+/- infinity
Vertical asymptotes
Explain with words & graphs what lim(x>+/- infinity) f(x)=L
Horizontal asymptotes.
Limit of f(x) is arbitrarily large & +/- provided x is sufficiently close to a (but not equal to a)
How to calculate limit
Easy, continuous function’s limits computed by substituting =limx->a f(x)=f(a)
Not nice (0/0, infinity) function’s limits computed by simplifying into nice function, finding holes, VA & HA asymptotes & finally substituting
How to find features of quotient functions
(x+3)/(x+3)(x-2) Hole: x=-3, Vertical A: x=2 sub in x=1.9, 2.1 to find +/- limits, Horizontal A: n<d y=0, n=d y=coe/coe, n>d y=oblique, sub in x=+/- infinity
Identify & explain what a function with a continuous domain is
f(x) is continuous at x=a if xaf(x)=f(x), unbroken, consistent graph, no abrupt changes in value (ex. Polynomials, trig, e^x, log(x))
Discontinuous (ex. piecewise, traffic light)
Points of Discontinuity
Jump Discontinuity: piecewise
Removable Discontinuity: hole
Infinity Discontinuity: vertical asymptotes
When given a function with paraments, how to select parameter values to make a function continuous
Set equations equal at x=a, if x<k=f(x), xk=g(x), f(k)=g(k), solve for variable
Average Rate of Change
Slope of a Secant Line =[f(x1)-f(x0)]/x1-x0