Spring 2018 Flashcards
What does it mean to say a limit is an indeterminate form? What are some examples?
A limit is referred to as an indeterminate form if the limits of its various parts do not provide a consensus about the overall limit.
Some examples are infinity/infinity, 0/0, 0*infinity, 1infinity
Is a limit of this form considered to be an indeterminate form (the limit cannot immediately be determined)? Why?
What methods can be used to compute a limit of this form?
Yes, the limit is an indeterminate form.
Since the numerator is growing without bound, this indicates the overall limit should also grow without bound.
However, the denominator growing without bound indicates the overall limit should be zero.
That these two ideas disagree is why this limit is an indeterminate form.
Some possible methods to compute this limit are either an application of L’Hopital’s Rule or re-expressing the function algebraically. It’s possible that, when rewritten, it won’t be indeterminate anymore.
Is a limit of this form considered to be an indeterminate form (the limit cannot immediately be determined)? Why?
What methods can be used to compute a limit of this form?
This limit is considered to be indeterminate.
When one term in a product is going to zero, it implies that the overall limit will also be zero. However, when one term in a product grows without bound, it implies the overall limit will also be infinity. That these limits disagree is why this limit is an indeterminate form.
A limit of this form usually requires re-writing the limit as a quotient (by dividing by the reciprocal of either term), which may then be in a form to which we can apply L’Hopital’s Rule.
(see: L’Hopital’s Rule I, II worksheet)
Is a limit of this form considered to be an indeterminate form (the limit cannot immediately be determined)? Why?
What methods can be used to compute a limit of this form?
Yes, the limit is an indeterminate form.
Since the numerator is going to zero, this indicates the overall limit should also go to zero.
However, the denominator going to zero indicates the overall expression should grow without bound.
That these two ideas disagree is why this limit is an indeterminate form.
Some possible methods to compute this limit are either an application of L’Hopital’s Rule or re-expressing the function algebraically. It’s possible that, when rewritten, it won’t be an indeterminate form anymore.
Is a limit of this form considered to be an indeterminate form (the limit cannot immediately be determined)? Why?
What methods can be used to compute a limit of this form?
Yes, the limit is an indeterminate form.
Since the base is going to 1, this indicates the overall limit should also go to one.
However, the exponent growing without bound implies the overall expression should grow without bound (or go to zero, if the base is less than 1).
That these two ideas disagree is why this limit is an indeterminate form.
Some possible methods to compute this limit are to rewrite the function using e^ln(function). This will allow us to turn what was an exponential function into a product, and possibly apply L’Hopital’s Rule by re-expressing that as a quotient.
(see: L’Hopital’s Rule III worksheet, PS19)
For which types of limits does L’Hopital’s Rule apply?
What does L’Hopital’s Rule assert?
L’Hopital’s Rule applies ONLY to “0/0” or “infinity/infinity” limits.
It says that if you encounter a limit that is an indeterminate form of either of these types, then the limit you are looking for shares is the same as the limit of the ratio of the derivatives, if the limit of the ratio of the derivatives is finite or infinite.
(re-expressed mathematically here)
Given:
an initial deposit
a nominal annual interest, compounded continuously
what function gives the described bank account’s balance as a function of time t, where t is in years?
Why?
B(t) = A*ert
where A is the initial deposit amount and r is the interest rate, expressed as a decimal (ex: 6% interest sets r=0.06)
WHY?
The general format for this function will always be A(1+r/n)nt,
where r is the annual interest rate and n is the number of compounding periods per year. To simulate continuous compounding, we want to compute the limit of this expression as n (the number of compounding periods) grows without bound. Since the limit of (1+r/n)n as n grows without bound is er , the limit of our entire expression will be Aert.
What characteristic might a limit have that would make you think it might be a good candidate for exponentiation/logarithmic algebraic rewriting, before applying L’Hopital’s Rule?
A limit might be a great candidate for this technique if it has been expressed as a complicated exponential, where the changing limit variable is in both the base and the exponent,
and the limit is an indeterminate form such as “1infinity”
What are some methods of evaluating limits?
Best case scenario: the behavior of the limit’s parts agree on how the overall limit should behave. This happens in forms of “0/infinity” or “infinity/0” “0*0” or “infinity * infinity”. Reasoning about the limit can be used in a situation like this.
If the limit is a “0/0” or “infinity/infinity” indeterminate form, it’s possible (especially with trig functions) that the limit may simplify if algebraically rewritten into something easier to compute. Otherwise, L’Hopital’s Rule may apply.
If the limit is a “0*infinity” limit, it may be algebraically rewritten as a quotient (by dividing by the reciprocal of either term), creating a form to which we may apply L’Hopital’s Rule.
If the limit is a “1infinity” limit, we may need to exponentiate in order to take the logarithm of our limit, allowing an exponential limit to be re-expressed as a product, which can be algebraically re-written, and may qualify for L’Hopital’s Rule to apply.
If the limit involves various growth terms (exponentials, polynomials, logarithms, etc.), we may be able to apply reasoning about which terms grow fastest in the numerator and denominator, allowing us to reason about the overall limit.
Given a picture of the graph of a rate function f(t),
how can you visualize the net change in amount over a given time interval?
Visually, the net change in the amount of the relevant quantity is represented by the signed area
between the curve and the t-axis over the given time interval.
Given a picture of the graph of a rate function f(t), how can we determine whether the net change in the relevant quantity is a net gain or loss over a particular time interval?
We look at the signed area between our function f(t) and the t-axis.
The area that lies above the t-axis signifies a gain in the amount of the relevant quantity.
The area that lies below the t-axis signifies a loss in the amount of the relevant quantity.
If the area above the t-axis is greater than the area below the t-axis over the given interval, then overall there has been a net gain.
If the reverse is the case (more area below the t-axis), then overall there has been a net loss.
Pictured is a visual example of net gain.
What does the following notation mean?
L6, R4, etc.
This notation refers to an approximating method for finding the area under a given curve on a given interval.
The subscript signifies how many equal length subintervals to break the interval into.
The area under the curve is then approximated by rectangles that span each subinterval, whose length is determined by either the function value on the right- or left-hand side of the interval, signified by the R or L, respectively.
(the attached is a picture of an R3 approximation of sin x from 0 to pi/2)
see: Worksheet and Problem Set on Net Change
Given ONLY the graph of a particular function f(t), under what circumstances can we compute the exact value of the definite integral?
When we are only given a picture of the function’s graph, we can compute the exact value when:
the signed area can be broken into geometric shapes whose area formulas we know
or the function on the interval has some symmetry such that the negatively and positively signed area cancel out, such as the integral of cos x from 0 to pi (same area above and below the x-axis!).
(picture of the former example attached)
What are three ways to think about the definite integral?
If we are looking at a picture of the function’s graph, it is the signed area between the curve and the t-axis.
If f is a rate function, this value is the net change in the relevant quantity over the interval [a,b].
And no matter what, the definite integral’s value is the limit of the left and right hand approximating methods as n, the number of subintervals, grows without bound (goes to infinity).
Under what circumstances are left- or right- hand approximations definitely over or underestimates for the value of a definite integral?
If a function is strictly increasing, then a right-hand appoximation will be an overestimate (the function value on the right side of each subinterval will be greater than that on the rest of the interval, making the rectangle overshoot the actual area).
In this case, the left-hand approximation will be an underestimate for the opposite reason (the function value on the left side of each subinterval will be less than the rest of the subinterval, making the rectangle underapproximate the area).
If the function is strictly decreasing, the opposite will be true: the right-hand approximation will underestimate the value of the integral (the function value on the right-hand side of each subinterval will be the least over the subinterval, making the rectangle underestimate the signed area), and the left-hand approximation will overestimate the value of the integral (the function value on the left-hand side in this case will be the greatest over the subinterval, making the rectangle overestimate the signed area).
If a function is both increasing and decreasing over the interval, we have to do a closer analysis. Be careful not to make assumptions in this more complicated case - it will depend on the function and its behavior over the interval!
Given a picture of the graph of an “arrival rate” function (i.e. the rate at which people arrive at a store/clinic/etc. as a function of time), and information about the maximum rate of service,
how can we identify when a line will form (value or interval for t), and how can we determine how long the line will grow to be (i.e. its maximum length)?
How is this related to the longest someone will be in line?
In problems like these, it is really important to continuously remind ourselves that we are looking at rate functions, not amount functions.
We can identify the times a line will form (and grow) as whenever the arrival rate is greater than the service rate. The first time value that happens will be when the line initially forms. The line will continue growing as long as the arrival rate lies above the service rate. If all we are given is the picture of the graph, this will usually be an approximation.
When determining the line’s maximum length, it is important to focus ONLY on the time interval during which the line is growing (t interval during which arrival rate > service rate). On this interval, the definite integral of the arrival rate will be the total number of people who arrive during this period, while the definite integral of the service rate will be the total number of people served during this period. In order to find the maximum length of the line, we want to compute the difference between the # of people who arrive and the # of people who are served from when the line starts growing to when the line stops growing. That is, we want to take the definite integral of the arrival rate - the definite integral of the service rate ONLY over the time interval when the line is growing (when arrival rate < service rate).
The attached is an example where the maximum rate of service is 30 people/hr, making line formation begin at t=10 and the line grows until t=14. The area of the unshaded blue region indicates the maximum line length.
- *The person who stands in line the longest is the person who gets in line when it is at its greatest length.** This may seem redundant, but think about that for a moment! If someone gets in line before this point in time, the line is still growing, and there are people who will be waiting in an even longer line behind them. If a person gets in line after this point in time, they are now getting into a line that is shorter (since it is shrinking) and won’t have to wait as long as the poor person who got in line when it was at its longest.
- *We can compute this max wait time by calculating the line’s maximum length (from the previous discussion) and dividing by the service rate.**
Think about how to approximate the value of a definite integral using left and right-hand sums. What is the procedure for this? What notation do we as a mathematical community use? Define every term used (define, meaning express with a mathematical equation) and which part of the approximation it is involved in.
In order to construct either a right-hand or left-hand sum approximation with n subintervals (Rn or Ln) for the integral of f(t) dt on the interval [a,b], we first want to make sure to break up the interval into n subintervals of equal width. In order to do this, we take the length of the interval and divide it by n to determine the length of each subinterval.
We give this distance a name, “delta x,” and it is equal to (b-a)/n, reflecting the process we just described.
We now need to partition the interval into n subintervals and name each endpoint of those subintervals. We define these with their variable name (usually x or t, but depends on the input of the function) and a serial number. The start of the first subinterval (x0) begins at the start of the larger interval at a and ends at x1 (at a+delta x). This process repeats over all n intervals, making the last subinterval’s end point xn, which is also the end of the larger interval, b.
For any value for k, xk=a+k*delta x.
For a right hand estimate, we use rectangles to approximate the area under the curve. Each rectangle’s width is delta x; and its height is determined by the function value on the right hand side of each subinterval.
For a left hand estimate, the rectangles’ width remains the same, but each height is determined by the function value on the left hand side of each subinterval.
One way to write the general term for sigma notation for each approximation method is f(xk)delta x. For the right hand sum, k will range from 1 to n; for the left hand sum, it will range from 0 to n-1.
Given a desired error and a strictly increasing or decreasing function, how can you determine the appropriate number of subintervals for a given level of accuracy when approximating the value of a definite integral?
When looking at left and right hand estimates for the definite integral of an increasing or decreasing function, one approximating method will provide an over estimate and one will provide an underestimate. Since the actual value of the integral will fall between these two approximations, the difference between them will be larger than the error of either one.
This means that |Ln-Rn| will be greater than the actual error of either approximation method. If we make |Ln-Rn| less than our desired error, it will guarantee our actual error is even smaller (and therefore reflects the level of accuracy we want).
When reflecting on the value of |Ln-Rn|, we notice that any rectangle whose height comes from an endpoint in the interior of the overall interval appears in both the left and right hand approximations (see attached picture). As a result, when computing |Ln-Rn|, most of the terms will cancel, except for the very first left hand rectangle f(x0)*delta x and the very last right hand rectangle f(xn)*delta x
|Ln-Rn|=|f(x0)-f(xn)|*delta x
Once all pieces are written in terms of their definitions (both the function values at x0 and xn, as well as delta x), this can now be set less than the desired accuracy, and the required n – the only remaining variable – can be solved for.
Our final statement should be n > some number, meaning we need at least this many rectangles to achieve the desired level of accuracy.
Bring to mind as many properties of the definite integral as possible. How can you make sense of them using the various perspectives of the definite integral?
What is the value of the definite integral after swapping the order of the bounds?
What is the sum of two integrals (of the same function) over intervals that share an endpoint?
What is the definite integral of the sum of two functions?
What is the definite integral of a scalar multiple of a function?
What are some skills that are useful in simplifying integrals like these?
Pictured are the major properties of the definite integral.
Swapping the order of the bounds changes the sign. This makes sense through a net change perspective - what is a gain over one time interval is a loss over the reverse time interval.
Net change also helps to makes sense of combining two neighboring intervals into one (e.g. net change from 1-2p +net change from 2-3p is the same as net change from 1-3p).
The definition of the definite integral helps to make sense of the integral of the sum of two functions. The sum of what was once individual rectangle heights will now be the height of rectangles for the functions’ sum. Similarly, multiplying a function by a scalar will make any individual rectangle k times its original height, the same height as the rectangles involved in the definition of the definite integral for the function kf(x).
In computing integrals involving these properties, it is important to remember that we can also compute the values of definite integrals when the signed area takes on geometric shapes.
Which is greater:
the integral of the absolute value of f
OR
the absolute value of the integral?
WHY?
The integral of the absolute value of f!
Taking the absolute value of the integral only guarantees that the final result is positive after the integral has been computed. During this process, positive and negative signed areas will cancel, in part, leaving only some of their magnitude left in the final answer.
When looking at the absolute value of the function first, then taking the definite integral, we are now working with an everywhere positive function, whose integral value will only increase over the interval.
Attached are two pictures, showing the regions that will cancel from being oppositely signed in the absolute value of the integral, versus the all positive areas of the integral of the absolute value of the function.
