+ Most people's favorite version of a linear equation is this:
+
+ Graph the following lines on the graphs paper below. + +
+
+
+
+
+ \begin{beforesolutions} +
++ \end{beforesolutions} +
+
+ The following equations all define a line,
+ but are not in the usual slope-intercept form,
+ i.e.
+ Turn the following equations into slope-intercept, and then graph them below + +
+
+
+
+
+
+
+
+
+
+ In some problems the quickest way to write a linear equation is like
+ this
+
+ Find the point-slope form equation of the line through the point
+ Turn the equation from (a) into slope-intercept form. +
+
+ Find the point-slope form equation of the line through the point
+ Turn the equation from (c) into slope-intercept form. \handoutfill +
++
+
+
+
+
+ This example is meant to show that sometimes it makes sense to think about a problem using the point-slope form of a line. +
+ ++ Suppose that today my son is 52 inches tall and growing at 1.5 inches + per year. + +
+ Roughly speaking, how tall will he be tomorrow? +
++ How tall will he be in one year? +
++ How tall will he be in two years? +
+
+ Write a formula for
+
+ Basically, tomorrow he'll be about the same height as today:
+
+ In one year he will grow roughly another
+ In two years he should grow another
+
+ This example is meant to show that it's actually quite easy to graph a + line in point-slope form. + +
+ On the graph paper below,
+ graph the point
+ Add to the graph a second large point, \tikz\draw[fill=black] circle (2pt);
+ that is
+ Draw a line through the two points you have labeled. +
+
+ Describe what the graph you made has to do with the line
+ The first lesson is that it was easy to draw this line geometrically:
+ draw one point, count over and up, draw a second point.
+ The second lesson is that we can see this information in the equation
and the
is the point we start with.
+ And the slope
+ is pretty much where we always see it.
+ So really, this shouldn't be any harder than using the
+ Add the fractions, and simplify if possible:
+
+ Add the fractions, and simplify if possible:
+
+ Get a common denominator and combine the fractions:
+
+ Get a common denominator and combine the fractions:
+
+ Get a common denominator and combine the fractions:
+
+ Get a common denominator and combine the fractions:
+
+ Multiply the fractions, and simplify if possible:
+
+ Multiply the fractions, and simplify if possible:
+
+ Multiply the fractions, and simplify if possible:
+
+ Simplify until you get a single fraction,
+ with no compound fractions:
+
+
+
+
+
+
+
+
+ There are two ways you can do this.
+ Multiply, then cancel:
+
+
+ As before, you can multiply then cancel:
+
+ I'll do this one in a fair amount of detail:
+
+ Recall:
+
+ Using the above properties, simplify the following. + +
+
+
+
+
+
+
+
+
+
+ Simplify the following
+
+ Challenge Problem:
+ Simplify the following
+
+
+
+ There's more than one order you can do this in,
+ and it really
+ doesn't matter too much which way you go.
+ But I think it does help
+ to get some sort of a strategy and try to follow that.
+ For
+ instance,
+ you could say I'll work from the inside out, and
+ simplify as I go.
Or you could say,
+ I'll work from the outside
+ in, and simplify at the end.
But what you probably
+ shouldn't
+ say is I'll randomly combine the inside and the
+ outside,
+ and move everything around until I think of something to do
+ with it.
+
+ I'll work from the inside out and simplify as I go:
+
+ Recall that
+ Simplify the following:
+ Simplify the following:
+
+ Simplify the following:
+
+
+
+
+ Simplify the following:
+
+ Simplify the following:
+
+ The main step is FOILfoiling
means you take
+ each thing on the left, and distribute it across the pieces on the
+ right.
+ The letters stand for First Outer Inner Last.
+ We apply FOIL to both
+ A quadratic function has the following form
+
+ Match the following quadratics with their graphs: see if you can do + this without using your calculator. + +
+
+
+
+
+
+ \begin{beforesolutions} +
+ +
+ {
+ {
+ {
+ {
+ {
+ \end{beforesolutions} +
+
+
+
+
+
+
+ Factoring a quadratic means to write it as a product.
+ Usually you
+ shouldn't bother to factor a quadratic unless the
+ There are various tricks in finding
and
+
and say add
+ I'm also including negative numbers in there.
+
+
+ Factor and solve
+ Factor and solve
+ Factor and solve
+ Factor and solve
+ Factor and solve
+ Factor and solve
+
+
+
+
+
+
+
+ A lot of times it's not worth factoring a quadratic, or it may be
+ impossible.
+ In these cases, just use the quadratic formula
+
+ Apply the quadratic formula to
+ Apply the quadratic formula to
+
+
+
+
+
+
+
+
+ In the formula
+ we simply replace each
+ with
+ With a little practice,
+ you should be able to find
+ For
+ is just a letter we use to stand for
+ something else.
+ If you get too fixated on
+ As in the last part, let's first write
+ The historic Senator Theater is the nearest movie theater to Loyola.
+ Their ticket prices for adults (non-students) seeing a 3D movie are
+
+ Basically we're just taking the given information about pricing and
+ times and arranging it using the notation for piecewise functions.
+ We put the prices first, and then times second. (In general we put
+ formulas first, and the conditions second.) In this case we get
+
+ Since July 1, 2016, the Maryland minimum wage is
+ Make up an integer
+ Make up an integer
+ Make up a non-integer
+ Make up a non-integer
+
+ (e) Time and a half rate is
+ What are the domains of these functions? + +
+
+
+
+
+ The domain is the set of all numbers that we can plug into our formula.
+ When I say we can plug in
+ I mean so that the result is defined, or calculate-able.
+ The way we find the domain is to look for the numbers that are not in it,
+ the numbers that make things undefined or un-calculate-able.
+ In other words, we look for the problem spots.
+
+ With algebraic functions like these there are only two possible + kinds of problem spots: division by 0, and square roots of negative + numbers. + These are problems, and so these are not in the domain. + +
+ For
+ For
+ For
+ For
+ For the two points
+ There are probably three useful ways to think about the slope
+
+ Note that all three of these formulas say the same thing:
+ rise
really means
+ is just shorthand notation for
+ change of
.
+
+ The main things you have to watch out for in this kind of problem are: (1) make sure you put the
+ Now we finish our calculation:
+
+ Find an equation of the line that passes through
+ We'll start with the point-slope form of the equation of a line:
+
+ From Example 1, we know the slope already,
+ There's nothing wrong with leaving the equation like this,
+ but most of
+ us are more used to the slope-intercept form, so we can distribute the
+ numbers and simplify:
+
+ A cab company has an initial charge of
+
+ There are three things to identify here:
+ where do we put the
+ The initial
+ means at the very beginning, in this case,
+ before we start driving at all.
+ This means it's the value when we've gone
+ The per
is a clue that this is the slope:
+ slope is a ratio, and per
always means ratio too.
+
+ Putting these together we should have
+
+ The correct somethings are
+ is our variable here, not slope
+ (since
+ ACME company has seen a decline in sales of their product. + In 2010 + they sold 28.4 million, while in 2016 they sold 22.7 million. + +
+ Find a formula for annual sales
+ Predict the sales in 2019. +
++
+ We have two points of the line:
+
+ Use
+ Table 1.14 shows world bicycle production (from \url{http://www.earth-policy.org/indicators/C48/}, accessed April 19, 2005.) Table 1.14 + World bicycle production, in millions +
++ Find the change in bicycle production between 1950 and 2000. + Give units. +
++ Find the average rate of change in bicycle production between 1950 and 2000. + Give units and interpret your answer in terms of bicycle production. +
++
+ We simply subtract the bicycle production during one year from
+ the production during another year:
+ million bicycles
,
+ so the full answer is 90 million
+ bicycles.
+
+ The average rate of change is, by definition,
+
+ Find the average rate of change of
+
+ The average rate of change is, by definition,
+
+ In this case that means
+
+ Consider two situations: (1) A company has an increase in sales from
+
+ Which absolute change is bigger? +
++ Which relative change is bigger? + Justify your answer. +
++
+ The change is situation (1) is
+ The relative change is situation (1) is
+
+ A company producing jigsaw puzzles has fixed costs of
+ Find formulas for the cost function, + the revenue function, and the profit function. +
+
+ Sketch a graph of
+ What is the marginal cost? +
++
+
+ The graph is shown below
+
+ The marginal cost is
+ The demand curve for a product is given by
+ At a price of
+ Find the equilibrium price and quantity. + Does your answer to part (a) support the observation that market forces tend to push prices closer to the equilibrium price? +
++
+
+
+ The gross domestic product,
+ 3
+ 8 billion dollars per year +
++
+
+ This describes a constant rate of change, so it is
+ linear.
+
+ For school year 2013
+ Assuming that the tuition continues to grow at the same rate,
+ what will it be for the 2019
+ We model the tuition with the following equation
+
+ For 2019 we have
+ For school year 2013
+ Find
+ We want tuition to fit into the formula
+
+ We plug the data into this equation:
+
+ Solve
+
+ We can't solve this problem all the way in our heads,
+ but you should be able to make a guess that the solution will be between
+ Here's how we solve it algebraically:
+
+ In the school year 2013
+ We plug this data into the model
+
+ So, the actual year should be about
+ A city's population starts at 600,000 in 2010 and has a continuous
+ growth rate of 5
+ We model this city with
+
+ A population, currently 200,
+ is growing at 5
+ Write a formula for the population,
+ Graph
+ Estimate the population 10 years from now. +
++ Find the doubling time of the population algebraically. +
++ Model the same population using a continuous growth rate, + compare the graph of this model with the graph from part (b). +
++
+
+
+
+ We solve for when
+ They are asking to change this model to the natural exponential
+ (base
+ Find the future value in 8 years of a
+ Find the present value of a
+
+ We'll use the equation
+
+ We'll use the equation
+
+ Figure 2.12 shows the cost,
+
+ { Figure 2.12} +
+ +
+
+ Is the average rate of change of the cost greater between
+ Is the instantaneous rate of change of the cost of producing
+ What are the units of these rates of change? +
++
+ In the graph below I've marked, by hand,
+ two line segments on the curve,
+ i.e.
+ In the graph below I've marked,
+ by hand, two tangent lines.
+ One is tangent at
+ The units are, as always,
+
+ Match the points labeled on the curve in Figure 2.14 with the given slopes. +
++ { Figure 2.14} +
+ +
+
+ We'll justify below the following answers:
+
+ Let's start by thinking about which point has negative slope:
+ only two points are marked where the graph is going down:
+
+ Now let's think about which point has
+ Finally, let's look at three remaining points,
+ where the slope is positive.
+ We can order them by how steep the graph is: It's steepest at
+ Suppose we drop a penny from the roof of a very tall building.
+ Then
+ the distance fallen is given by
+
+ Find the average velocity from
+ Estimate the instantaneous velocity at
+
+
+ To estimate the instantaneous velocity we find the average velocity over
+ shorter and shorter time intervals around
+ Using a calculator or an equivalent app,
+ estimate
+ By definition we have
+
+ The way to estimate this is to calculate
+
+ The best way to get a range of values like this is to make a table,
+ either in your calculator, or in something like \url{www.desmos.com}:
+
+ Once we have this information,
+ we should look just around the rows
+ that have
+ Use Figure 2.16 to fill in the blanks in the following statements
+ about the function
+
+
+ { Figure 2.16} +
+ +
+
+
+ When we write something like
+
we mean that when
+
+ When we write something like
+ Match the functions in Problems 18
+ \afterpage{} +
+
+ Although we have to do all four functions,
+ we don't have to do them in order.
+ So, we'll start with the simplest one:
+ The function in
+ Now let's look at
+ Now let,s look at
+ Which graph has
+ Finally, let's look at
+ Write the Leibniz notation for the derivative of the given function + and include units. + +
+ The cost,
+ An employee's pay,
+
+ Since
+ Since
+ An economist is interested in how the price of a certain item
+ affects its sales.
+ At a price of
+
+
+
+ In problems like these,
+ the best way to explain the meaning
is to write a complete,
+ correct, English sentence that uses the least amount of technical jargon as possible.
+ In this case, here are some examples:
+ When the price is
+ If we price it at
+ A price of
+ We'll sell 2000 items at a price of
+
+ It may help here to practice writing this in Leibniz notation:
+ items per dollar
and the rate of
+ change is If the price changes to
+
+
+ At the price of
+
+ The cost,
+
+
+
+ This means that it costs
+ This means it will cost about
+ For the function
+ Use a table of numbers to approximate
+ Using linear approximation and your answer to part (a),
+ to approximate
+
+ Here are some of the values we look at for the difference
+ quotient:
+
+ The basic idea here is that we can use the tangent line to
+ approximate values on the original graph.
+ Thus
+
+ The area of Brazil's rain forest,
+
+ Interpret
+ Find and interpret the relative rate of change of
+
+
+
+ The world population in billions is predicted to be approximately
+
+ By definition,
+
+ But to use this we first need to find
+ We use this with
+ Now we can finish this problem:
+
+ Another way to put this is that the relative rate of change in
+ At which of the labeled points, if any,
+ are both
+ The only point where they are both positive is
+ Here's the proof.
+ Recall that
+ Next, recall that
+ or +
+
+ So, the only points where
+
+ Give the signs of the first and second derivatives for the following functions. + Each derivative is either positive everywhere, + zero everywhere, or negative everywhere. +
++
+ Since this graph is increasing,
+ we have
+ Since this graph is decreasing,
+ we have
+ Since this graph is increasing,
+ we have
+ The temperature outside on a given day is given by
+ This poll should be done through Poll Everywhere and then discussed + online. + +
+ The temperature was below freezing but getting warmer. +
++ The temperature was below freezing and getting colder. +
++ We do not no whether the temperature was above or below freezing. +
++ The temperature was higher at noon than at 6 AM. +
++ The temperature was lower at noon than at 6 AM. +
++ The temperature was rising but at a slower rate as the morning progressed. +
++ The temperature was rising but at a faster rate as the morning progressed. +
++ The temperature was falling and at a faster rate as the morning progressed. +
++ The temperature was falling but at a slower rate as the morning progressed. +
+
+ There is not an official
solution to this,
+ because it is meant to be a discussion.
+
+ Let
+ The price of the stock is falling faster and faster.
+
+ The price of the stock is getting close to its peak,
+ at which it will remain for a little while.
+
+ The price of the stock is skyrocketing.
+
+
+ The phrase price
means faster and faster.
Probably the middle one.
+ In that case,
+ it's concave down (concave down is part of a frown
) and so
+ For the price to be getting close to its peak,
+ we need that the graph is increasing,
+ so
+ The price skyrocketing means that it is increasing,
+ so Concave up is part of a cup
) and so
+ It costs
+ We don't have enough information to solve this exactly but we can
+ approximate it:
+
+ Figure 2.55 shows a total cost function,
+ { Figure 2.55} + +
+ Estimate the marginal cost when the production level is 20 and interpret it. +
+
+ Is the marginal cost greater at
+ Is the marginal cost greater at
+
+ The best way to do this graphically is to put a ruler on the graph at
+ By looking at the tangent lines when
+ By looking at the tangent lines when
+ Figure 2.57 shows part of the graph of cost and revenue for a car + manufacturer. + Which is greater, marginal cost or marginal revenue, + at + +
+
+
+
+ { Figure 2.57} +
+
+ Remember not to look at the gap between the two lines,
+ or at which line is higher at which point.
+ Rather, look at the slopes of the two lines,
+ at the points
+ To produce 2000 items, the total cost is
+ 2001 items +
++ 1999 items +
++ 2050 items +
++
+ With marginal cost being
+ With marginal cost being
+ With marginal cost being
+ Cost and revenue functions for a charter bus company are shown in Figure 2.58.
+ Should the company add a 50th bus?
+ How about a 90th?
+ Explain your answers using marginal revenue and marginal cost.
+ { Figure 2.58} +
+
+ We need to look at the difference between the marginal costs and marginal revenues.
+ At
+ So the additional revenue of adding the 50th bus will be greater than the additional cost. + So yes, it should add a 50th bus. +
+ +
+ At
+ Let
+ With practice, you can probably just write this down in one step:
+
+ But when you are still learning these steps, it might help to break it
+ down more:
+
+ For
+ For
+
+ Before taking the derivative of
+ Before taking the derivative of one over
.
+
+ Now we can take the derivative:
+
+ For
+ For
+
+
+ In this case we should clean up
before
+ you take even the first derivative:
+
+ Let
+ Find the equation of the tangent line to
+ Find when
+
+ We will fill in the following equation
+
+ First we rewrite
+ the question, at least in our heads:
+
+ Let
+ Find
+ Find the equation of the tangent line at
+ Compare the graph of
+
+ Using the constant multiple rule, sum rule, and exponential
+ rules:
+
+
+ We show
+ The human population of the entire world can be modeled by
+
+ Find the estimated rate of growth in 2020, + and interpret your answer, with units. +
+
+ Recall that
+
+ Also, the year 2020 means that
+ Here is a simple interpretation: +
+ +++ +In 2020 the population will increase by 0.083 + billion people each year.+
+ Note: A simpler, and better, + statement would convert 0.083 billion to 83 million. + I didn't do that above just because I didn't want to introduce an extra calculation or step into the solution. +
+
+ Find the marginal revenue function if
+ Basically marginal revenue
+ means we should take the derivative of
+
+ Now we apply these rules, as well as the power rule from earlier:
+
+ Suppose we are given
+ The main idea here is to apply the chain rule to inside
of the
+ exponent, we should use the chain rule.
+ We start with the basic
+ rule (do nothing in this case),
+ we don't change the inside, and then
+ we multiply by the derivative of the inside.
+ I'll write this in two
+ ways: (1) using an extra variable,
+
+ Now we combine this with the rest of the formula for
+ Find the derivatives of + +
+
+
+ For Chain Rule problems,
+ we want to think of what is the
+ outside function
+ and what is the inside function
,
+ and we need each of these to be simple enough that we know how to find its derivatives.
+ Sometimes it is best to think of the
+ outside function
+ as the last calculation you do.
+
+ For (a), we see that the last calculation that would be done would
+ be to take the fifth-power what is in the parentheses.
+ Thus we'd
+ have
+ For (b), we need to rewrite the function, changing the
+ radical in the denominator to a negative exponent:
+
+ We can clean this up, but please be clear in your head:
+ we are done
+ with the derivative at this point.
+ The rest is algebra which
+ might be useful, but you also might sometimes not want to do
+ it, and in any case it's not the new part here.
+
+ Find the derivatives of + +
+
+
+ With exponential functions it's a little tricky to spot what
+ inside function
means.
+ Basically, we should just learn with
+ experience that for these, inside
means what's on top.
+ You can
+ make this a little easier to see if you realize that the exponents
+ are basically inside implied parentheses:
+
+ Now inside
really means inside,
+ it's inside the parentheses.
+ If
+ you combine this with the Chain Rule you get the following pattern
+
+ We'll use this in both parts below. +
+ +
+ For (a), we use the Chain Rule on each of the terms, using the pattern
+ above.
+ Thus we get
+
+ For (b), we use the pattern for Exponential Functions and Chain Rule Combined:
+
+ Note: it's important that you include the parentheses around
+ Find the derivative of
+ Here we have: outside =
+ This gives us the pattern for natural logarithms with the Chain Rule:
+
+ Use the Product Rule to differentiate
+ When you are first learning the product rule, you may need to take a
+ couple of extra steps to keep the notation straight: (1) Write the
+ product rule formula on your paper, right in your solution.
+ If you
+ write this every time, I promise that you'll be able to remember it
+ better, and you'll be able to use it better. (2) Label the parts of
+ the function with
+ It's not really necessary to do anything else to this answer:
+ it's not going to simplify much,
+ and we weren't asked to factor it or anything.
+ It is traditional to put that last 5
+ in front of the
+ Let me show you how you might solve this problem if you've been doing
+ product rules for a while.
+ In that case, you might not label derivative of the
+ 1st, times the second, plus the 1st times the derivative of the
+ second.
+
+ That may seem like a lot to keep track of, or do in your head, + but if you just write and say one part at a time, it's not so bad. + In any case, + do not feel like you should try to do more in your head; + feel free to label everything and arrange all the parts like we did in our first solution above. + Just know, that if you want to, + you can start to label less as you go. +
++ Find the derivatives of the following: + +
+
+
+ Both of these involve the Product Rule. +
+ +
+ For (a), the first is
+ Again, you probably shouldn't try to
+ simplify
any farther here.
+ We've taken the derivative, and we're done with that;
+ unless there's a reason to factor it, just stop.
+
+ For (b), in addition to the Power Rule and Product Rule,
+ we need to
+ remember that
+ Use the Quotient Rule to differentiate
+ When you are first learning the Quotient Rule, you may need to take
+ a couple of extra steps to keep the notation straight: (1) Write the
+ Quotient Rule formula on your paper, right in your solution.
+ If you
+ write this every time, I promise that you'll be able to remember it
+ better, and you'll be able to use it better. (2) Label the parts of
+ the function with
+ As before, just leave it alone and don't try to simplify,
+ unless you've been asked to,
+ or if there's a reason to.
+ In other words,
+ simplify at your own risk. (In case you want to practice,
+ see if you can get
+ We can also learn to take the quotient rule without labelling
+ everything, just like we talked about with the Product Rule.
+ Here's
+ how it looks.
+ Say the following words to yourself as you write things
+ down: derivative of the top,
+ times the bottom, minus the top, times
+ the derivative of the bottom,
+ all over the bottom squared.
+
+ There's a rhyming mnemonic for this as well:
+ low d-hi minus hi d-low, over the bottom squared, and away we go!
.
+ Here, low d-hi
+ means derivative of the top (d-hi) times the bottom, etc.
+
+ Let
+ Find
+ Find the equation of the tangent line at
+ For part (a),
+ we need to find
+ Note that we're using
+ For part (b), we'll fill in
+ (as always)
+ the point-slope equation of a line:
+
+ They didn't ask us for it, but just for kicks,
+ here's a graph of
+ Suppose
+
+
+ The point
+
+
+ The
+ Sketch the graph of a function with two local maxima and one local minimum. +
++ Sketch the graph of a function that has two critical points. + One should be a local maximum and one should be neither a local maximum nor local minimum. +
+
+ There is no solution
for this example:
+ we will discuss it as a group.
+
+ [First Derivative Test]
+ Suppose
+ If
+ If
+ If
+ [Second Derivative Test]
+ Suppose
+ If
+ If
+ If
+ Find all local extrema of the function below, using the Second
+ Derivative Test:
+
+ Recall the main steps we need for the Second Derivative Test: + +
+ Find
+ Find critical numbers (when
+ Find
+ Plug the critical numbers into
+ We start by finding
+ Now we find
+ Finally, we find the critical points,
+ i.e.
+ Thus the function has two critical points:
+
+ Find and classify all the critical points of the function
+
+ We need to find the critical numbers by first finding
+ Now we set it equal to
+ Note the common factors of
+ It would take a lot of work
+ (multiple Product Rules)
+ to find
+ So we look at
+ To organize my work, I draw a number line with the critical numbers marked.
+ Then I calculate
+ From the above information and the First Derivative Test,
+ we see that
+ there is a local maximum at
+
+ Finally, we can summarize our answer
+
+ An graph of
+ An inflection point is where
+ Find all critical points and inflection points of the function
+
+ Identify each critical point as a local max, local min, + or neither. +
+
+ We start by taking the derivative and setting it equal to
+ Our critical numbers are
+ Therefore we can identify the behavior at each critical number
+
+ Now we find the inflection points.
+ We start by setting the second
+ derivative equal to
+ It's easy to see that
+ Finally, we calculate the
+ Find all inflection points of each of the functions below
+
+ We start by finding where the second derivative of each of these is
+
+ Now we look to the left and the right of
+ In other words,
+
+ Now we look to the left and the right of
+ In other words,
+ Sketch a possible graph of
+ From the information in the chart,
+ we see that
+ Can we say if these points are local max or mins?
+ Yes.
+ We have that
+ This makes it easy to see that
+ From the information in the chart,
+ we see that
+ Can we say if inflection points really occur here?
+ Yes.
+ We have that
+ This makes it easy to see that
+ Graph a function with the given properties. + +
+ Has local minimum and global minimum at
+ Has local minimum at
+ Has no local or global maxima or minima. +
+
+ Has local and global minimum at
+ Discussion opportunity. +
+
+ [Global Max/Min Test]
+ To find the global max and global min of
+ Find the critical numbers +
+
+ Compare the values for
+ Values for
+ means you plug the critical numbers and ends of the interval into
+ For the function
+
+ We start, as always, by finding the critical numbers.
+ In other words,
+ we take the derivative and set it equal to
+ For
+ Now it's rather easy to finish:
+ we plug these critical numbers back
+ into
+ Now, simply look at the
+ We can summarize our findings:
+
+ The energy expended by a bird per day,
+
+ To find the minimum value of
+ Looking at the second derivative,
+ we see it is always positive,
+ so a foraging time of
+ Suppose you're looking at a demand curve.
+ What does the point on the curve where
+ Let
+ What is the meaning of
+ Describe in words how the marginal cost changes as the quantity produced increases. +
++ Explain the concavity of the graph (in terms of economics). +
++ Explain the economic significance + (in terms of marginal cost) + of the point at which the concavity changes. +
+
+ Do you expect the graph of
+
+
+
+ The marginal cost decreases slowly,
+ and then increases as quantity produced (
+ Concave down implies decreasing marginal cost, + while concave up implies increasing marginal cost. +
+
+ Since the concavity of the graph is determined by
+ Because of volume pricing of supplies, + etc., there are production levels in which the additional cost to produce more items + (marginal cost) + would decrease. + But at some point, + one would have to pay workers overtime, hire more workers, + rent more storage space, + add more delivery trucks, etc. + Thus at some point the additional cost to produce more + (marginal cost) + will go up. +
+
+ A demand function is
+ Find an expression for the total revenue
+ Find the marginal revenue,
+
+ Compare with the change in total revenue when production changes from
+ Before we start, make sure you understand the basic formula
+
+
+
+
+ The demand equation for a product is
+ We start by finding
+ Now, as always,
+ we take the derivative and set it equal to
+ This is a critical number.
+ To verify that it's a maximum,
+ note that the original function
+
+ To find the price that corresponds to this quantity we use the
+ original function for
+ Finally, we'll get the revenue that we expect at this quantity.
+ We
+ can calculate it in the simplest possible way: we've got
+ We could also plug
+ Production of an item has fixed costs of
+
+ The relationship between price,
+
+ Find the profit function
+ How many items should the company produce to maximize profit? (Give your answer to the nearest integer.) What is the profit at that production level? + What is the price charged at that production level? +
++
+
+ Be very carefully!
+ Some problems, many, use
+ As always,
+
+ Suppose you are making something, say T-shirts,
+ and you want to model how much revenue you'll bring in.
+ You know the demand curve of your T-shirts,
+ i.e. how many T-shirts you'll sell at a given price.
+ You can write the demand curve in two ways,
+ Which do you think makes more sense:
+ use
+ There is no solution
here:
+ it's an(other) opportunity to discuss something!
+
+ Figure 4.63 shows cost with
+ Find the average cost when the production level is 10,000 units and interpret it. +
++ Represent your answer to part (a) graphically. +
++ At approximately what production level is average cost minimized? +
+
+
+
+
+ Conceptually, our answer to part (a) can be viewed as the slope
+ of a line through
+ To minimize the average cost,
+ we would look at different lines from the origin to points on the curve,
+ and we would ask: Out of all such lines,
+ which one has the smallest possible slope?
+ Here I've drawn 4 of them, just to illustrate:
+
+ The cost function is
+ The marginal cost is simply the derivative
+
+ In other words,
+ marginal cost is constant, and so
+ The average cost
+ In other words,
+ the added cost of making one additional unit is about
+ The average cost per item to produce
+ What is the total cost,
+ What is the minimum marginal cost? + What is the practical interpretation of this result? +
++ At what production level is the average cost a minimum? + What is the lowest average cost? +
+
+ Compute the marginal cost at
+
+ Since
+ There are two derivatives here:
+
+ We will learn in this problem that there are two ways to
+ minimize average cost:
+ find a critical point of
+ We simply plug in analytical
reason.
+ What that means is that we will take the
+ derivative of
+ {4.6 Formulas for Elasticity}
+ Let
+ \item
+ \item
+ \item
+ The elasticity of the demand for eggs is
+ a
+ a
+ With
+ With With
+ By comparing the effect change in prices are on eggs (
+ In Fall 2013, the undergraduate enrollment at Loyola University
+ Maryland was
+ Will a
+ Can you find a way to predict this answer without repeating all the calculations? +
++
+ Since the elasticity is
+ We can realize percent change in
+ {4.6 Critical Points in Elasticity}
+ In general,
+ the elasticity determines whether
+ The demand function of Loyola T-shirts is
+ Find
+ Find
+ When
+
+ Never forget, revenue equals the the number of things you
+ sell, times how much you sell each one for:
+
+
+ There's a general rule for how
+ means that if
+ The odometer on our car is broken, + but we really need an estimate of how far we're driving. + The speedometer readings are shown below; + use them to estimate the distance traveled over the first 30 minutes. + Find a lower estimate and an upper estimate. +
+
+ To find the lower estimate, we look at each 10 minute interval, and
+ choose the lowest velocity that we see.
+ For example, from
+ To get our upper estimate, we repeat most of the above steps, but in
+ each 10 minute interval we use the highest velocity that we see.
+ We
+ get the following numbers in each interval:
+
+ The figure below shows the velocity,
+
+ Let's start by focusing on the main idea here: + area below this curve = total distance traveled. + So, what we are really + going to find is the area under the curve. + We haven't been told to use a particular method to estimate this area, + so we'll decide on our own how to go, + and we'll split it up into rectangles and triangles, as shown +
+
+ Now we simply estimate the heights of each rectangle and triangle
+ using the numbers on the grid,
+ and guessing when it's between two
+ numbers.
+ I won't spend time worring about exactly how well you read
+ the graph, but here are the numbers I found:
+
+ The velocity of a vehicle on a track is given by
+ The main idea here is to turn this back into area:
+
+ So, let's draw this graph,
+ and then calculate the distance under it (more precisely,
+ between the graph and the horizontal axis,
+ and between
+ Maybe you know the area formula for a trapezoid, but if not, + it's easy to break this shape up into a rectangle and a triangle: +
+
+ Now we can finish our calculation:
+
+ Using the graph of
+ Left-hand sum with
+ Right-hand sum with
+ Left-hand sum with
+ Right-hand sum with
+
+ Since
+ We'll be briefer this time:
+
+
+
+ Using the figure below (Figure 5.36 in the text), decide whether + each of the following definite integrals is positive or negative. + +
+
+
+
+
+
+ The main idea of this example is simply to translate integrals into net area.
+ So,
+ But, sooner or later it's useful to have different parts shaded in, + so I'll go ahead and do that +
+
+
+
+
+
+ The following graph shows the function
+
+
+
+
+
+ This example is similar to the previous example, + the only difference is that in the last example we could just estimate what was positive and what was negative, + but here we can calculate the exact integrals using basic geometric formulas for area. + We'll start by dividing the regions, + shading them and labeling them. +
+
+ This is labeled with (a) above.
+ It's a triangle plus a rectangle,
+
+ This is labeled with (b) above.
+ It's a triangle,
+ This is labeled with (c) above.
+ It's one quarter of a circle with radius
+
+
+ Explain in words what each integral represents and give the units + +
+
+
+
+
+ To figure out what any
+ integral is giving you, look at
+ everything after the integral sign:
+
+
+
+ The marginal cost
+ Estimate the total cost of making 60 T-shirts. +
++ What is the total variable cost of making 60 T-shirts? +
++ Estimate the difference in cost between making 60 T-shirts and 100. +
++
+
+ The total variable cost is the part of our previous answer
+ that didn't come from fixed cost.
+ In other words
+
+
+ A cup of coffee is put into a
+ Find
+ What is
+ What does
+ How much does the temperature change in the first 5 minutes in the room? +
++
+ Do you still remember how to take derivatives?
+ Basically, we
+ need the Chain Rule here for
+
+
+
+