Section 01: mathematics Review

Graphing Data

It is said that a photo is worth a thousands words. Likewise, a graph can frequently be offered to better understand relationships between variables. Recall the on a graph, there is a horizontal axis and also a upright axis.

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Along the horizontal axis, we begin at the origin or a zero. To the appropriate numbers increase and also to the left numbers decrease.

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On the upright axis, we additionally start at the origin or zero and also the numbers together we walk up increase and the numbers as we go under decrease.

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Combining the horizontal and also vertical axes allow us come look at the relationships in between variables in two-dimensional space. While we will at times use all four quadrants, we generally spend most of our time in the first quadrant or the space where both the horizontal and vertical variables are positive.

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Plotting clues on a Graph

If the miscellaneous values the x and y space given, the points can be plotted top top the graph.

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Once the points space plotted, the dots can be linked to kind a heat or curve.

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In our example, there is a straight relationship in between x and y. We can express the same information, utilizing an equation. The intercept worth is the worth of y once x is zero.

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In modeling causal relationships, the value of an endogenous or dependency variable counts upon the worth of other variables in the model. When graphing, we typically place the dependent variable on the upright or y-axis. Those variables whose worths are figured out outside the design are recognized as exogenous or independent variables. This are typically graphed ~ above the x or horizontal axis.

A confident or straight relationship in between the 2 variables occurs when the rise in the independent variable causes an increase in the dependence variable. For example, as revenue increases the amount world will consume also increases.

A negative or inverse relationship occurs as soon as the increase in one variable decreases the value of the various other variable. Together the price the a good falls, the variety of people ready to purchase the an excellent increases.

In ours equation, over there is a confident relationship in between x and also y. When x boosts by one, y increases by 2. The steep of a line is determined by taking the adjust in the upright amount split by the change in the horizontal amount. We will certainly let the Greek prize Delta stand for the change. In our example, together x increases by 2, y increases by 4 so the slope would certainly a confident 2.

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From an equation, we room able to derive the point out in a table by plugging in the worths of x and computing the value of y.

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Slope the a Curve

A curve will have various slopes at various points follow me the curve. By acquisition the change in y end the change in x, we room able to gain an calculation of the curve at the point. As shown in the table, the slope gets steeper as we relocate along the curve.

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If we compute the slope over a distance roughly three, we discover that the slope from 0 to 6 is 12 and also from 2 to 4 is additionally 12.

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If us were to attract a tangent line (a line that touches the curve just at one place) whereby x amounts to 3 we might then compute the steep of the tangent. In act so, us would uncover the steep is again 12.

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Marginal Analysis

One of the an essential tools in business economics is marginal analysis. As economist look in ~ relationships in between variables, they want to understand what will be the adjust in one variable offered a change in one more variable. As the readjust in x gets smaller and smaller, we find the gain the steep of the tangent. Thus, gift able come compute the slope of the tangent tells us what the marginal readjust is in the dependency variable.

Slope that the tangent = ΔY / ΔX together the readjust in x gets really small

One powerful mathematical device in calculus is referred to as the derivative. The derivative tells united state the steep of the curve in ~ a certain point. Although that is no a requirement for this class, an instance is noted for those who have actually previously had calculus. The derivative merely tells us the change in y given a very little change in x, or in other words the steep of the curve at a certain point. Offered the function, Y = 4 + 2x2, the first derivative gives us a steep of the tangent in ~ a offered point. To uncover the slope of the tangent as soon as x = 3, we plug in the value of 3 for x in the very first derivative and also find the slope is 12.

The derivative gives us the steep of the curve.

ΔY / ΔX = steep of the curve

Y = 4 + 2x2

First derivative = ΔY / ΔX = 4x

Slope of the tangent once x = 3 is 4(3) = 12

Increases in the intercept value shifts the curve upwards by the quantity of the rise in the intercept. In our example, the intercept value is enhanced from 2 come 4.

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Increases (decreases) in the slope reasons the curve to be steeper (flatter).

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Equilibrium

Often, we desire to discover an equilibrium whereby the one curve would certainly intersect through another, such similar to supply and also demand.

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We space able to uncover the market equilibrium either by graphing or algebraically. If on the need side, we are offered Price = 50 – 2 (Quantity Demanded) and on the supply side, we are provided Price = 10 + 2 (Quantity supplied). We can simply plugin values, plot the points, and also find where demand intersects supply.

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Even there is no graphing the curves, we are able to view that at a price that $30, the quantity demanded equates to the quantity supplied.

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If we graph it, we uncover that in ~ price of 30 dollars, the quantity offered would be 10 and also the amount demanded would be 10.

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Solving Algebraically

An different way, is to deal with the trouble algebraically. Due to the fact that at equilibrium the amount supplied equates to the amount demanded and also the price will be the same for both. We are able to set the two equations equal and also solve.

P = 50 – 2Qd and also P = 10 + 2 Qs

Set the two equations equal

50 – 2 Q = 10 + 2 Q

Our first step is to acquire the Qs together, by including 2Q come both sides. ~ above the left hand side, the an unfavorable 2Q to add 2Q cancel each various other out, and on the appropriate side 2 Q plus 2Q offers us 4Q.

P = 50 – 2Qd and also P = 10 + 2 Qs

Set the two equations equal

50 – 2 Q = 10 + 2 Q

+ 2Q + 2Q

50 = 10 + 4Q

Our next step is to acquire the Q through itself. We can subtract 10 from both sides and also are left through 40 = 4Q. The last step is to divide both political parties by 4, which leaves us through an equilibrium quantity of 10.

Set the two equations equal

50 – 2 Q = 10 + 2 Q

+ 2Q + 2Q

50 = 10 + 4Q

-10 = -10

40 = 4Q

40/4 = 4/4 Q

10 = Q

Our next step is to get the Q by itself. We can subtract 10 indigenous both sides and are left through 40 = 4Q. The last action is to divide both sides by 4, which pipeline us v an equilibrium quantity of 10.

Equilibrium Q = 10

P = 50 – 2Qd or ns = 10 + 2 Qs

P = 50 – 2(10) or ns = 10 + 2(10)

P = $30 or ns = $30

Determining the Optimums

Math as a tool enables economists to determine the optimums. Because that example, where earnings are maximized or costs are minimized.

Finding the slope of a curve deserve to be extremely valuable when determining the maximum or minimum. Because that example, let’s say a agency uses labor to create a product, called widgets. The extr or marginal output from each added worker originally increases, then decreases together the other resources come to be relatively much more scarce.

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If the company wants to discover how many workers will maximize their output, they would certainly look in ~ the suggest where the slope of the curve goes come zero. In our example if the firm wanted come maximize output, it would usage 8 employees to execute so.

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Charts and also Graphs

Charts and also graphs are typically used to display information. A pie chart shows the part or family member magnitude the the complete amount the is consisted of by each of the components.


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Because that example, these pie charts display screen the sources of federal government revenues and also the main areas of federal government expenditures.