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
Solve for f’(1), if f(x)=x^2
2
Linear Approximation + Use
Linear Approximation/Tangent Line to f(x) at x=a is L(x)=f(a)+f’(a)(x-a), used to approximate hard values of functions
Linear approximation to f(x)=e^x at x=a=0, f’(0)=1
Find e^1/10
L(x)=f’(0)+f’(0)(x-0)=e0+1(x-0) = 1+x
L(1/10)=1+1/10=1.1
Euleur’s Number e
lim(h>0) eh-1/h=1
Solve for derivative of e^x
e^x
Differential Equation + 2 Examples
Differential Equation y’(t)=y(t): a unknown function & its derivatives with a solution that satisfies it
- y=e^x, y’=e^x, velocity vs acceleration, compound interest, bacteria growth, population growth
- y=Ce^x, y’=Ce^x, h>0(Ce(x+h)-Ce^x)/h=h>0(Ce^xeh-Ce^x)/h=limh>0 Ce^x(eh-1)/h=Cex
Instantaneous Rate of Change
Derivative of f(x) at x0 is f’(x0)=d/dxf(x)=lim(h>0)[f(x+h)-f(x)]/h
Doesn’t exist at corners, sudden changes in slope, jumps or where the limit is infinite number
Linearity of Differentiation + Proof
Proof using the limit definition of derivative & use to break down derivatives of sums + constant multiples
Power Rule + Proof
Proof using the limit definition of derivative & use for integer exponents
Derivative of a function
f’(x) = df/dx =limh>0(f(x+h)-f(x))/h
Used to solve derivatives, show how derivatives can be equivalent, etc.
Find derivative of f(x)=3x+x^2
f’(x)=3+2x
Derivative of an exponent (a^x) & tan(x)
d/dx 4^x=4^xlog(4)
d/dx a^x=a^xlog(a)
d/dx a^f(x)=a^f(x)log(f(x))f’(x)
d/dx tan(x)=sec^2(x)
Reciprocal Rule + Proof
See notes
Product Rule + Proof
See notes
Find derivative of f(x)=x^2+1/x^2
f’(x)=2x-2/x^3
Find derivative of f(x)=x*e^x
f’(x)=e^x+x*e^x
Quotient Rule + Proof
See notes
Find derivative of f(x)=(x^3+1)(x^2+2)
f’(x)=3x(x^2+2)+(x^3+1)2x
Graphically show trig f(x)s & their derivatives
See notes
Addition Formulas
sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
limh>0 sin(h)/h=1
limh>0 cos(h)-1/h=0
Derivative sinx, cosx & tanx proof
See notes
Chain Rule
d/dxf(g(x)) = f’(g(x))g’(x) = df/dgdg/dx
Differentiate f(x)=sin(3x)
cos(3x)(3)
Differentiate f(x)=(x^3-2x+1)^6
6(x^3-2x+1)^5(3x*2)
Implicit Differentiation
- Taking derivative of both sides using Chain rule
- Solve for dy/dx
Differentiate x^2+y^2=1, then find x where y’=1 using Algebra & Diagram
dy/dx=-2x/2y
x=+/- 1/sqrt(2)
Logarithmic Differentiation
Applying the natural logarithm to both sides of equation to make it easier to differentiate, especially in f(x)^g(x), combining Chain Rule+Implicit Differentiation+Logarithmic Rules
- Take log of both sides, simplifying using log rules
- Take d/dx of both sides (implicit differentiation+chain rule)
- Isolate for y’(x)
Differentiate x^a using logarithmic differentiation
f(x)= x^a
logf(x) = logx^a
1/f(x) * f’(x) = alogx
1/(x^a) * f’(x)=a/x
f’(x) = a/x(x^a)
f’(x)= ax^(a-1)
How to solve a related rate question?
- Draw & label a picture + Write down what is known
- Write down what you wish to find
- Write down an equation that relates the thing whose rate y (reduce to 1 variable)
- Differentiate & Solve
- Check if answer makes sense
Rate of Change of Area = L’W + LW’
Rate of Change of Focal Length of Lense
Rate of Change of radius of expanding circle
Rate of change of water level entering inverted cone
Rate of Change of angle of a moving person across a room