Continuity and Asymptotes
There are 4 reasons a function is discontinous.
Name them
1 - divide by zero
2 - piecewise functions
3 - ln of zero or a negative
4 - square root of a negative
When taking the derivative or integral of an equation like 1/(x²), what would you need to do first? What about for ∜(x^5)
Change terms with powers so that it is x^( ). A power in the denominator will become negative. Roots will turn into fractional powers.
How do you find relative extrema?
To find relative extrema for f(x):
1. find critical numbers
2. sign chart testing intervals around critical numbers
3. if f'(x) changes from (-) to (+) → min, or if f'(x) changes from (+) to (-) → max
4. See if asking what or when
5. answer
6. justifications: relative max b/c f'(x) changes from (+) to (-) OR relative min b/c f'(x) changes from (-) to (+) at that value
How do you know that a problem is a related rate? What are the other words for rates?
It is a good chance a problem is a related rate when the word “when” or “at the instant” is in the problem. You know for sure when there is more than one rate. Any word that means something is changing is a rate (like moving, or shrinking, or increasing or such). The steps to related rates are
What are all the relationships between velocity, acceleration, and position (for both derivatives and integrals)?
p '' = v ' = a
p ' = v
∫∫a = ∫v = p
∫a = v
In area and volume problems, when you see the word x-axis cross it out and rewrite what? When you see the word y-axis cross it out and rewrite what?
When you see x-axis cross it out with y = 0.
When you see y-axis cross it out and write x = 0.
How do you know if piecewise function is continuous or not?
To find continuity of a piecewise function check to see if the heights match at the boundary. If they do not, then the boundary is a discontinuity point.
How do you take the derivative of two equations being multiplied together?
How do you take the derivative of two equations being divided?
How do you take the derivative of an equation that is inside of another equation?
To take the derivative of two equations being multiplied use the Product Rule:
d/dx[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)
To take the derivative of two equations being divided use the Quotient Rule:
d/dx[f(x)/g(x)]=[f'(x)g(x)-f(x)g'(x)]/[(g(x))^2]
To take the derivative of an equation that is inside of another equation use the Chain Rule:
d/dx[f(u)]=f'(u)•u'
How do you find points of inflection?
Do a sign chart or look at given information to find any of the following
- f''(x) change in sign
- f'(x) change in slope
What are the steps you should follow when trying to find integrals?
The steps to integration are:
1. u-sub
2. arcs?
3. simplify
How do you know if a particle changes directions? How do you know if a particle is moving to the left or moving to the right?
A particle changes direction if you do a sign chart for velocity and the velocity changes signs. You know a particle is moving left if the velocity is negative and moving right if the velocity is positive.
What are the steps to find the area between two graphs?
To find the area between two graphs...
1. Sketch the graphs and shade the region R (graph the graph if not graphed for you)
2. find a and b (when the graphs intersect) and label the a and b on the graph
3. label top equation and bottom equation on the graph
4. take the integral ₐ∫ᵇ top − bottom dx
Of the 4 reasons for discontinuity (divide by zero, piecewise functions, ln of 0 or negative, square root of negative) WHICH ONE IS A REMOVEABLE DISCONTINUITY?
Divide by zero is the only removable discontinuity.
If you plug the zero from the bottom into the top and get zero then you have a hole/removable discontinuity. If you plug the zero on the bottom into the top and get anything other than zero then you have a vertical asymptote/non-removable discontinuity.
What are the two limit definitions of the derivative?
What is each one used for?
Limit of the difference quotient
lim_{h->0} (f(x+h)-f(x))/(h)
lim h→0 [(f(x+h)-f(x))/h]
and the alternative form
lim_{x->#} (f(#)-f(x))/(x-#)
The limit of the difference quotient will be on multiple choice and will need to be recognized. The alternative form is used to define differentiability.
What is the second derivative test? How does it work?
The second derivative test is a shortcut to find relative extrema.
f''(x) < 0 means concave down and relative max
f''(x) > 0 means concave up and relative min
What are the steps to finding a particular solution of a differential equation?
The steps to find a particular solution:
1. get y’s with dy’s and x’s with dx’s
2. integrate
3. put the +C with x’s
4. solve for y (if you e the plus C, it becomes times C) (stop here for general solutions)
5. plug in the initial condition to solve for the C
6. find the domain
How do you know if a particle changes which side it is on? How do you know if a particle is on the right or on the left?
A particle changes which side it is on if you do a sign chart for position and the position changes signs. You know a particle is on the right if position is positive and you know it is on the left if the position is negative.
What method do you use to find the volume of a figure rotated around y = #? What are the steps?
This is called Washers method. To find the volume of a figure rotated around y = #:
1. Sketch the graph (if not already done for you), shade the area, and label the revolution line.
2. Find a and b and label on the graph.
3. Label the graph that is farthest from the revolution line and the graph that is closest to the revolution line.
4. find R(x) and r(x)
R(x) : using only the graph that is farthest from the revolution line and the revolution line itself: R(x) = the higher – lower one
r(x) : using only the graph that is closest to the revolution line and the revolution line itself: r(x) = the higher – lower one
5. Find the integral:
pi int_a^b(R(x)^2-r(x)^2)dx
Graphically and Mathematically:
What is a jump discontinuity?
What is an infinite discontinuity?
What is an essential discontinuity?
Jump: heights do not match from left and right. Happens with absolute value center or piecewise
Infinite: Line goes vertical . Equation = #/0.
Essential: sin(#/0) cos(#/0), aka sin(∞) or cos(∞) which would be oscillating
How do you find the derivative of an equation that has x and y on the same side?
Use implicit differentiation. To implicit differentiate:
1. Take the derivative of each piece
2. Put dy/dx when take derivative of a y.
3. Get dy/dx terms on one side and the other stuff on the other side.
4. Factor out dy/dx (if more than one)
5. Divide the dy/dx stuff to the other side.
How do you find the equation of a tangent line?
How is this different than when a problem says to find the
slope of the tangent line?
Use y-[y₁]=[m](x-[x₁])
or y=[m](x-[x₁])+y₁.
x₁ is given, find y1 by substituting x₁ into original equation, m is slope of the line (derivative at x₁).
This is different than asking for the slope of the tangent line because the slope is just the derivative value at x₁ and the equation is the full y-[y₁]=[m](x-[x₁]) .
How do you know that a problem is asking you to find the particular solution?
A problem is asking you to find the particular solution if a differential equation is given (dy/dx=equation) and it asks for what the original equation or solution was.
How do you find speed and total distance?
Speed = |velocity| =
Total Distance = ₐ∫ᵇ speed dt = ₐ∫ᵇ |velocity| dt
What are the steps to find the volume of a figure with known cross sections?
To find the volume of a figure with known cross
1. Sketch the graphs and shade the region R (graph the graph if not graphed for you)
2. find a and b (when the graphs intersect) and label the a and b on the graph
3. label top equation and bottom equation on the graph
4. plug (top – bottom) into s for A(s)
5. Find the integral:
ₐ∫ᵇA(x)dx =
What does a limit tell you about a graph?
What are the steps to finding limits?
How do you know if a limit is undefined?
A limit tells you what height (y-value) that a graph is approaching as the graph approaches the x value you are taking the limit at, or you can talk about it is the height that two roller coasters collide at or not.
Steps to limits: 1-plug in the number, 2 – l’hopitals rule 3-table (using your calculator). A limit is undefined if the y-values approach different numbers in your table.
A limit is undefined if the y-values approach different numbers in your table.
What are the different words for saying the derivative equals zero?
Horiztonal Tangent Line
Critical Number
How do you approximate f(#)? How do you know if this approximation is an over or an under approximation?
To approximate f(#), just plug # into the equation of the tangent line, substitute for the x value. Remember eqn of tan. line: y=[m](x-[x₁])+y₁
ex) Approximation of f(2.1) = f'(2)([2.1]-2)+f(2)
What are the steps to related rates?
SREDWU:
1. Sketch (only if it doesn’t give a shape in the problem)
2. Find the Rates
3. Make an Equation (only involving the letters from the rates)
4. Take the Derivative of that equation
5. Plug in the When
6. Find the Units
How do you find if a particle moving towards the origin or away from the origin?
To find if the particle is moving towards the origin or away do a double sign chart for position and velocity and if they match then the particle is moving away from the origin and if they are opposite then they are moving towards the origin. (You really need to understand why these work and not just memorize it)
How do you know if a right or left hand Riemann sum is an over or under approximation?
If the function is increasing then the right hand is an over approximation and the left hand is an under approximation.
If the function is decreasing then the right hand approximation is an underapproximation and the left hand is an overapproximation.
(Don’t just memorize this, know that it involves increasing and decreasing and be able to draw it).
How do you find horizontal asymptotes?
How do you find limits as x approaches infinity?
What is the order of functions from slowest to fastest?
You find a horizontal asymptote by finding the limit as x approaches infinity and negative infinity.
You find the limit as x approaches infinity by comparing the top and bottom of an equation (if the top is faster then you get infinity or undefined, if the bottom is faster then you get zero, and if they match then you get their coefficients).
The order of functions from slowest to fastest: trig, constant, lnx, squareroot, linear, polynomial, exponential
What is the average rate of change and the instantaeous rate of change on a graph?
The average rate of change is the same as the slope of the secant line which gives the slope from one point to another on the graph.
The instantaneous rate of change is the same as the slope of the tangent line which gives the slope of one point on the graph.
What are the justifications for increasing, decreasing, relative maximum, relative minimum, relative maximum, point of inflection, concave up, concave down?
increasing = 𝑓’(𝑥) > 0
Decreasing = 𝑓’(𝑥) < 0
Relative maximum = 𝑓’(𝑥) changes from positive to negative there Relative minimum = 𝑓’(𝑥) changes from negative to positive there Point of inflection = 𝑓”(𝑥) changes signs there
Concave up = 𝑓”(𝑥) > 0
Concave down = 𝑓”(𝑥) < 0
In Differential Equation FRQs, if you are asked to find the particular solution what two things do you need to solve for at the end of the process?
Solve for "y"
Solve for "C"
How do you know if an objects speed is increasing or decreasing?
To find if speed is increasing or decreasing you need to do a double sign chart with velocity and acceleration and if they match signs then speed is increasing and if they are opposite signs then the speed is decreasing. (You really need to understand why these work and not just memorize it)
How do you know if a trapezoidal approximation is an over or under approximation?
If the graph is concave up then the trapezoidal approximation is an overapproximation. If the graph is concave down then the trapezoidal approximation is an underapproximation. (Don’t just memorize this, just know that it involves concavity and draw the scenario).