Economic Applications of Single-Variable Calculus Derivatives in Economics

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In economics, derivatives play a crucial role in analyzing various economic phenomena such as marginal amounts, maximization, minimization, graphing, elasticity, and growth. This involves understanding derivatives of single-variable functions, slopes, instantaneous slopes, and the applications of derivatives in economic contexts. Derivatives help in determining the impact of small changes in variables, which is essential for decision-making in economics.

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2. Economic Applications of Single- Variable Calculus Derivative Origins Single Variable Derivatives Economic uses of Derivatives 1

2. Economic Applications of Single- Variable Calculus 2.1 Derivatives of Single- Variable Functions 2.2 Applications using Derivatives 2

2. Economic Applications of Single-Variable Calculus In economics, derivatives are used in various ways: Marginal amounts (slope) Maximization Minimization Graphing Elasticity and Growth 3

2.1 Derivatives of Single-Variable Functions Slope: -The impact on the y variable when the x variable increases by 1 -consider the following graph -what is the impact of changing how much time you spend on assignment 1? 4

What is the impact of one more hour? Slope = rise/run Slope AB (2->3 hours) =(y1-y0)/(x1-x0) =(100-35)/(1-3) = y/ x = (y1-y0)/(x1-x0) =-32.5 mistakes Assignment 1 Mistakes Slope BC (3->4 hours) =(y1-y0)/(x1-x0) =(20-35)/(5-3) =-7.5 mistakes 120.00 100.00 80.00 Mistakes 60.00 40.00 But what is the impact of a SMALL time change at 3 hours? 20.00 0.00 1 2 3 4 5 6 7 8 9 10 Hours Spent on Assignment 1 5

2.1 Derivatives of Single-Variable Functions --in order to find the slope AT B (the impact of a small time change), one must find an INSTANTANEOUS SLOPE -slope of a tangent line at B -derivative at B 6

2.1 Tangents Slopes 120.00 100.00 80.00 Quantity 60.00 Series1 40.00 20.00 0.00 1 2 3 4 5 6 7 8 9 10 Price The green tangent line represents the instantaneous slope 7

2.1 Instantaneous Slope To calculate an instantaneous slope/find a derivative (using calculus), you need: 1) A function 2) A continuous function 3) A smooth continuous function 8

2.1.1 A Function Definitions: -A function is any rule that assigns a maximum and minimum of one value to a range of another value -ie y=f(x) assigns one value (y) to each x -note that the same y can apply to many x s, but each x has only one y -ie: y=x1/2 is not a function x = argument of the function (domain of function) f(x) or y = range of function 9

Function: y=0+2sin(2pi*x/14)+2cos(2pi*x/14) Each X 4 3 2 Has 1 Y 1 0 x y -1 1 3 5 7 9 11 13 15 17 19 -2 -3 -4 x 10

Not a Function: Here each x Corresponds to 2 y values Often called the straight line Often called the straight line test test 11

2.1.1 Continuous -if a function f(x) draws close to one finite number L for all values of x as x draws closer to but does not equal a, we say: lim f(x) = L x-> a A function is continuous iff (if and only iff) i) f(x) exists at x=a ii) Lim f(x) exists x->a iii) Lim f(x) = f(a) x->a 12

2.1.1 Limits and Continuity In other words: i) The point must exist ii) Points before and after must exist iii)These points must all be joined Or simply: The graph can be drawn without lifting one s pencil. 13

2.1.2 Smooth -in order for a derivative to exist, a function must be continuous and smooth (have only one tangent) Slopes 40.00 35.00 30.00 25.00 Quantity 20.00 Series1 15.00 10.00 5.00 0.00 1 2 3 4 5 6 7 8 9 10 Price 14

2.1.2 Derivatives -if a derivative exists, it can be expressed in many different forms: a)dy/dx b)df(x)/dx c)f (x) d)Fx(x) e)y 15

2.1.2 Derivatives and Limits -a derivative (instantaneous slope) is derived using limits: ) ( ' x f + ( ) ( ) f x h f x = lim h ( ) h 0 This method is known as differentiation by first principles, and determines the slope between A and B as AB collapses to a point (A) x=0+2sin(2pi*t/14)+2cos(2pi*t/14) 4 3 2 1 0 x x -1 1 3 5 7 9 11 13 15 17 19 -2 -3 -4 t 16

2.1.2 Rules of Derivatives -although first principles always work, the following rules are more economical: 1) Constant Rule If f(x)=k (k is a constant), f (x) = 0 2) General Rule If f(x) = ax+b (a and b are constants) f (x) = a 17

2.1.2 Examples of Derivatives 1) Constant Rule If f(x)=27 f (x) = 0 2) General Rule If f(x) = 3x+12 f (x) = 3 18

2.1.2 Rules of Derivatives 3) Power Rule If f(x) = kxn, f (x) = nkxn-1 4) Addition Rule If f(x) = g(x) + h(x), f (x) = g (x) + h (x) 19

2.1.2 Examples of Derivatives 3) Power Rule If f(x) = -9x7, f (x) = 7(-9)x7-1 =-63x6 4) Addition Rule If f(x) = 32x -9x2 f (x) = 32-18x 20

2.1.2 Rules of Derivatives 5) Product Rule If f(x) =g(x)h(x), f (x) = g (x)h(x) + h (x)g(x) -order doesn t matter 6) Quotient Rule If f(x) =g(x)/h(x), f (x) = {g (x)h(x)-h (x)g(x)}/{h(x)2} -order matters -derived from product rule (implicit derivative) 21

2.1.2 Rules of Derivatives 5) Product Rule If f(x) =(12x+6)x3 f (x) = 12x3 + (12x+6)3x2 = 48x3 + 18x2 6) Quotient Rule If f(x) =(12x+1)/x2 f (x) = {12x2 (12x+1)2x}/x4 = [-12x2-2x]/x4 = [-12x-2]/x3 22

2.1.2 Rules of Derivatives 7) Power Function Rule If f(x) = [g(x)]n, f (x) = n[g(x)]n-1g (x) -work from the outside in -special case of the chain rule 8) Chain Rule If f(x) = f(g(x)), let y=f(u) and u=g(x), then dy/dx = dy/du X du/dx 23

2.1.2 Rules of Derivatives 7) Power Function Rule If f(x) = [3x+12]4, f (x) = 4[3x+12]33 = 12[3x+12]3 8) Chain Rule If f(x) = (6x2+2x)3 , let y=u3 and u=6x2+2x, dy/dx = dy/du X du/dx = 3u2(12x+2) = 3(6x2+2x)2(12x+2) 24

2.1.2 More Exciting Derivatives 1) Inverses If f(x) = 1/x= x-1, f (x) = -x-2=-1/x2 1b) Inverses and the Chain Rule If f(x) = 1/g(x)= g(x)-1, f (x) = -g(x)-2g (x)=-1/g(x)2g (x) 25

2.1.2 More Exciting Derivatives 2) Natural Logs If y=ln(x), y = 1/x -chain rule may apply If y=ln(x2) y = (1/x2)2x = 2/x 26

2.1.2 More Exciting Derivatives 3) Trig. Functions If y = sin (x), y = cos(x) If y = cos(x) y = -sin(x) -We see this relationship graphically: 27

2.1.2 More Derivatives Reminder: derivatives reflect slope: Sine(blue) and Cosine(red) 1.5 1 sin(x),cos(x) 0.5 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 -0.5 -1 -1.5 x 28

2.1.2 More Derivatives 3b) Trig. Functions Chain Rule If y = sin2 (3x+2), y = 2sin(3x+2)cos(3x+2)3 Exercises: y=ln(2sin(x) -2cos2(x-1/x)) y=sin3(3x+2)ln(4x-7/x3)5 y=ln([3x+4]sin(x)) / cos(12xln(x)) 29

2.1.2 More Derivatives 4) Exponents If y = bx y = bxln(b) Therefore If y = ex y = ex 30

2.1.2 More Derivatives 4b) Exponents and chain rule If y = bkx y = bkxln(b)k Or more generally: If y = bg(x) y = bg(x)ln(b) X dg(x)/dx 31

2.1.2 More Derivatives 4b) Exponents and chain rule If y = 52x y = 52xln(5)2 Or more complicated: If y = 5sin(x) y = 5sin(x)ln(5) * cos(x) 32

2.1.2.1 Higher Order Derivatives -First order derivates (y ), show us the slope of a graph -Second order derivatives measure the instantaneous change in y , or the slope of the slope -or the change in the slope: -(Higher-order derivates are also possible) 33

2.1.2.1 Second Derivatives Here the slope increases as t increases, transitioning from a negative slope to a positive slope. x=15-10t+t*t 10 5 0 x x 1 2 3 4 5 6 7 8 -5 -10 -15 t A second derivative would be positive, and confirm a minimum point on the graph. 34

2.1.2.1 Second Derivatives Here, the slope moves from positive to negative, decreasing over time. x=15+10t-t*t 50 40 30 x x 20 10 0 1 2 3 4 5 6 7 8 t A second derivative would be negative and indicate a maximum point on the graph. 35

2.1.2.1 Second Order Derivatives To take a second order derivative: 1) Apply derivative rules to a function 2) Simplify if possible 3) Apply derivative rules to the answer to (1) Second order differentiation can be shown a variety of ways: a)d2y/dx2 c)f (x) e) y b) d2f(x)/dx2 d) fxx(x) 36

2.1.2.1 Second Derivative Examples y=12x3+2x+11 y =36x2+2 y =72x y=sin(x2) y =cos(x2)2x y =-sin(x2)2x(2x)+cos(x2)2 y =-cos(x2)2x(4x2)-sin(x2)8x-sin(x2)2x(2) =-cos(x2)8x3-sin(x2)12x 37

2.1.2.2 Implicit Differentiation So far we ve examined cases where our function is expressed: y=f(x) ie: y=7x+9x2-14 Yet often equations are expressed: 14=7x+9x2-y Which requires implicit differentiation. -In this case, y can be isolated. Often, this is not the case 38

2.1.2.2 Implicit Differentiation Rules 1) Take the derivative of EACH term on both sides. 2) Differentiate y as you would x, except that every time you differentiate y, you obtain dy/dx (or y ) Ie: 14=7x+9x2-y d(14)/dx=d(7x)/dx+d(9x2)/dx-dy/dx 0 = 7 + 18x y y =7+18x 39

2.1.2.2 Implicit Differentiation Examples Sometimes isolating y requires algebra: xy=15+x y+xy =0+1 xy =1-y y =(1-y)/x 40

2.1.2.2 Implicit Differentiation Examples x2-2xy+y2=1 d(x2)/dx+d(2xy)/dx+d(y2)/dx=d1/dx 2x-2y-2xy +2yy =0 y (2y-2x)=2y-2x y =(2y-2x) / (2y-2x) y =1 41

2.1.2.2 Implicit Differentiation Examples Using the implicit form has advantages: 3x+7y8=18 3+56y7y =0 56y7y =-3 y =-3/56y7 vrs. y=[(18-3x)/7)1/8 y =1/8 * [(18-3x)/7)-7/8 * 1/7 * (-3) Which simplifies to the above. 42

2.2.1 Derivative Applications - Graphs Derivatives can be used to sketch functions: First Derivative: -First derivative indicates slope -if y >0, function slopes upwards -if y <0, function slopes downwards -if y =0, function is horizontal -slope may change over time -doesn t give shape of graph 43

2.2.1 Positive Slope Graphs Linear, Quadratic, and Lin-Log Graphs 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 44

2.2.1 Derivative Applications - Graphs Next, shape/concavity must be determined Second Derivative: -Second derivative indicates concavity -if y >0, slope is increasing (convex) -if y <0, slope is decreasing (concave, like a hill or a cave) -if y =0, slope is constant (or an inflection point occurs, see later) 45

2.2.1 Sample Graphs x = 2, slope is increasing; graph is convex x=15-10t+t*t 10 5 0 x x 1 2 3 4 5 6 7 8 -5 -10 -15 t 46

2.2.1 Sample Graphs x =-2, slope is decreasing; graph is concave x=15+10t-t*t 50 40 30 x x 20 10 0 1 2 3 4 5 6 7 8 t 47

2.2.1 Derivative Applications - Graphs Maxima/minima can aid in drawing graphs Maximum Point: If 1) f(a) =0, and 2) f(a) <0, -graph has a maximum point (peak) at x=a Minimum Point: If 1) f(a) =0, and 2) f(a) >0, -graph has a minimum point (valley) at x=a 48