MAT 4810: Calculus for Elementary and Middle School Teachers MAT 4810: Calculus

Computing for Elementary and Middle School Teachers Derivatives & Integrals Part I

Charles Delman Computing Derivatives & Integrals

The Definition Part I & Meaning of the Derivative Function

Local Charles Delman Coordinates

Computing Derivatives of More Complicated October 17, 2016 Functions Believe You Can Solve Problems Without Being Told How!

MAT 4810: Calculus for Elementary and Middle School Teachers I would like to share the following words from the introduction to Don Cohen’s book. I hope you will take the attitude they Computing Derivatives & express to heart and share it with your own students. Integrals Part I Charles “You can do it! You must tell yourself that. Don’t think Delman because you haven’t done a problem before, that you can’t do The Definition & Meaning of it, or that someone else must show you how to do it first (a the Derivative Function myth some people want to use to keep others in ignorance).

Local You can do it! Don’t be afraid. You’ve learned how to do the Coordinates hardest things you’ll ever do, walking and talking – mostly by Computing Derivatives of yourself. This stuff is much easier!” More Complicated Functions Recall: Linear Rates of Change

MAT 4810: Calculus for Elementary and Middle School Teachers mi To travel at 60 miles per hour ( h ) is to travel a distance Computing Derivatives & of 60∆t miles in ∆t hours: ∆d = 60∆t, where d is Integrals Part I distance in miles and t is time in hours; ∆d and ∆t are

Charles the changes in these quantities, respectively. Delman A rate such as miles per hour − that is, a ratio of the The Definition & Meaning of change in output over the change in input of a function − the Derivative Function gives a purely linear relationship between these changes. Local That is, the change in output is simply a multiple of the Coordinates change in input. Computing Derivatives of More Complicated Functions Recall: Non-Linear Rates of Change

MAT 4810: Calculus for Elementary and Middle For most functions, the relationship between a change in School Teachers input and the resulting change in output is not linear. Computing Derivatives & This is because the ratio between these changes is not Integrals Part I constant: the change in output is not a constant multiple

Charles of the change in input. Delman When you go on a road trip, you don’t drive at the same The Definition & Meaning of velocity all the time. Your average velocity over different the Derivative Function intervals of time will vary. Local Only those functions whose graphs are straight lines Coordinates exhibit a constant linear relationship between any change Computing Derivatives of in input and the resulting change in output. The graphs of More Complicated most functions are are curved. Functions The Equation of a Line

MAT 4810: Calculus for Elementary and Middle (x,y) School Teachers

Δy = y - y0 Computing

Derivatives & (x0,y0) Δx = x - x0 Integrals Origin for Δx and Δy: Part I (Δx, Δx) = (0,0) Charles Delman

The Definition & Meaning of the Derivative Given a fixed real number m, some initial input value x0, Function and corresponding output value y0 = f (x0): Local Coordinates ∆y y − y Computing 0 Derivatives of = m ⇔ = m ⇔ y − y0 = m(x − x0). More ∆x x − x0 Complicated Functions Constant rate of change is the defining property of those functions whose graphs are straight lines. Linear Functions are Central − But Not Enough

MAT 4810: Calculus for Elementary and Middle School Linear functions are relatively simple to understand and Teachers have many useful properties. Computing Derivatives & There is a whole field of mathematics called linear algebra. Integrals Part I But, as even the example of a road trip illustrates, linear Charles Delman functions alone are not adequate for modeling reality or fully developing the capacity of mathematics and science. The Definition & Meaning of the Derivative To take advantage of the properties of linear functions, we Function study more complicated functions using linear functions. Local Coordinates We do this by studying the linear relationship between Computing changes in input and output at each point where such a Derivatives of More relationship applies. Complicated Functions Recall: The Definition of the Derivative Function

MAT 4810: Calculus Let y = f (x) define a function f . for Elementary and Middle 0 School The definition of the derivative function, f , is Teachers

Computing 0 ∆y f (x + ∆x) − f (x) Derivatives & f (x) = lim = lim Integrals ∆x→0 ∆x ∆x→0 ∆x Part I if this limit exists. Charles Delman ∆y ∆x will not necessarily approach a limit as ∆x → 0. If The Definition f (x+∆x)−f (x) & Meaning of lim∆x→0 ∆x exists, f is differentiable at x. the Derivative 0 Function This definition is based on the idea that f (x) represents Local the instantaneous rate of change of f at input x. Coordinates 0 Computing In other words, f (x) gives the linear relationship between Derivatives of changes in input and output at point (x, y). More Complicated Functions In general, this rate will vary from point to point. Let’s look at this graphically. The Slope of a Curve

MAT 4810: 0 Calculus The value of the derivative f (x) gives the slope of the for Elementary and Middle curve y = f (x) at the point (x, y). School Teachers

Computing Δy f(x+Δx)-f(x) Derivatives & = Integrals Δx Δx Part I

Charles (x+Δx, f(x+Δx) Delman (x, f(x))

The Definition & Meaning of the Derivative Function Δx

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Local x x+Δx Coordinates

Computing Derivatives of 0 More If f (x) is defined, the curve is “smooth” at (x, f (x)). Complicated Functions Under increasing magnification, it would look straighter and straighter. Examples & a Pattern

MAT 4810: Calculus for Elementary and Middle Using this definition, we were able to calculate the School derivatives of the following functions: Teachers 0 Computing If f (x) = c, where c is a constant number, then f (x) = 0. Derivatives & Integrals The derivative of the identity function i, defined by Part I i(x) = x, is given by i 0(x) = 1. Charles If f (x) = x 2, then f 0(x) = 2x. Delman If f (x) = x 3, then f 0(x) = 3x 2. −1 0 −2 The Definition If f (x) = x , then f (x) = −x . & Meaning of the Derivative Function A clear pattern emerges, leading us to the following

Local conjecture: Coordinates n 0 n−1 Computing If f (x) = x , where n is an integer, then f (x) = nx . Derivatives of More We will verify this conjecture shortly, but first let’s look Complicated Functions graphically at these functions and their derivatives. Local Coordinates: The (∆x, ∆y)-Coordinate System

MAT 4810: Calculus for Elementary Since the slope of the curve y = f (x) generally varies from and Middle School point to point, it is helpful to view the curve in local Teachers coordinates based at each point (x, y) = (x, f (x)). Computing Derivatives & By local coordinates, we mean that we place the origin at Integrals Part I (x, y) = (x, f (x)). Charles We have already considered the (∆x, ∆y)-coordinate Delman system, in which we graph ∆y = f (x + ∆x) − f (x). (Note The Definition & Meaning of that x is fixed in this equation.) the Derivative Function Thus, ∆x denotes change in input away from the value x Local and ∆y denotes the corresponding change in the output Coordinates of f away from the value y = f (x). Computing Derivatives of More The graph ∆y = f (x + ∆x) − f (x) is just a translation of Complicated Functions the graph of f taking a fixed value (x, y) = (x, f (x)) to the origin. The Graph ∆y = f (x + ∆x) − f (x)

MAT 4810: Calculus for Elementary and Middle School Teachers

Computing Δy Derivatives & Integrals Part I

Charles Delman Δy = f(x+Δx) - f(x) The Definition & Meaning of the Derivative (x,y) = (x, f(x)) Function Δx Local Coordinates

Computing Derivatives of More Complicated Functions Local Differential Coordinates: The (dx, dy)-Coordinate System

MAT 4810: Calculus for Elementary and Middle School Teachers When we want to look at the slope of the curve y = f (x)

Computing at a particular point (x, y) = (x, f (x)), it is helpful to Derivatives & Integrals graph the tangent line to the curve at (x, y). Part I The tangent line is simply the straight line passing Charles 0 Delman through (x, y) with slope f (x).

The Definition Thus, the slope of the tangent line matches the slope of & Meaning of the Derivative the curve at (x, y). Function

Local We use local coordinates (dx, dy) to graph the tangent Coordinates line function, which is defined by dy = f 0(x)dx. (Note Computing 0 Derivatives of that x, and hence f (x), is fixed in this equation.) More Complicated Functions 0 The Graph dy = mx · dx = f (x)dx

MAT 4810: Calculus for Elementary and Middle School dy Teachers dy = mx⋅dx = f'(x)dx Computing Derivatives & Integrals Part I

Charles Delman (x,y) = (x, f(x)) dx The Definition & Meaning of the Derivative Function

Local Coordinates Thus, dx denotes a change in input away from the value x Computing and dy denotes the corresponding linear change in output Derivatives of More that applies at that point. Complicated Functions dy is called the differential change in f resulting from a differential change in input dx from x. Recall: Algebraic Combinations of Functions

MAT 4810: Calculus for Elementary We create new functions by combining functions and Middle 2 School algebraically. For example, if f (x) = x , g(x) = 5, and Teachers i(x) = x: Computing Derivatives & We can define the function f + i by the definition Integrals 2 Part I (f + i)(x) = f (x) + i(x) = x + x.

Charles We can define the function gi = 5i by the definition Delman gi(x) = 5i(x) = 5x.

The Definition We can define the function gf = 5f by the definition & Meaning of gf (x) = 5f (x) = 5x 2. the Derivative Function We can define the function 5fi by the definition 2 3 Local (5fi)(x) = 5(x )(x) = 5x . Coordinates We can define the function f + gi = f + 5i by the Computing 2 Derivatives of definition (f + 5i)(x) = f (x) + 5i(x) = x + 5x. More 1 We can define the function f by the definition Complicated 1
1 1 −2 1 Functions f (x) = f (x) = x2 = x . Note that the domain of f does not include 0. Complicated Derivatives from Simpler Ones

MAT 4810: Calculus for Elementary and Middle Rules have been developed for computing derivatives of School Teachers algebraic combinations of functions.

Computing Derivatives & The purpose of these rules is to compute the derivatives of Integrals Part I functions using the derivatives of simpler ones. Charles Our emphasis will be on understanding why the rules Delman work. Therefore, we will restrict our attention to the sum The Definition & Meaning of rule and the product rule, which are sufficient to compute the Derivative enough derivatives to explore some applications. Function Local For example, since we know the derivatives of the Coordinates functions defined by f (x) = x2, g(x) = x3, and h(x) = 5, Computing Derivatives of these rules will allow us to compute the derivative of the More 2 3 5 Complicated function defined by j(x) = 5x + x + x . Functions The Rules for Derivatives are Not So Simple!

MAT 4810: Calculus for Elementary and Middle 0 School Warning! The formula that defines f (x) is the limit Teachers of a somewhat complicated function of ∆x based on Computing Derivatives & the values of f at the two inputs x and x + ∆x: Integrals Part I 0 f (x + ∆x) − f (x) Charles f (x) = lim Delman ∆x→0 ∆x

The Definition Therefore, we cannot expect the derivatives of & Meaning of the Derivative algebraic combinations to depend on the derivatives of Function the functions being combined in a naively simple Local Coordinates manner − and they don’t! Computing Derivatives of In particular, the derivative of a product is not the product More Complicated of the derivatives. Functions Summary of Useful Notation

MAT 4810: Calculus for Elementary Let y = f (x) and z = g(x). Then, in local coordinates: and Middle School ∆y = f (x + ∆x) − f (x) and ∆z = g(x + ∆x) − g(x). Teachers 0 ∆y 0 ∆z Computing Thus f (x) = lim∆x→0 ∆x and g (x) = lim∆x→0 ∆x ; and Derivatives & Integrals Please remember that an expression such as ∆x is a single Part I variable representing a single number. They two symbols Charles Delman (such as ∆ and x) cannot be separated.

The Definition Think of ∆x as “the change in x”. Clearly, & Meaning of 2 2 2 the Derivative (the change in x) is not (the change in) times x ! Function

Local Similarly ∆y (the change in y), dx (the differential change Coordinates in x), and dy (the differential change in y) are single Computing Derivatives of variables that cannot be separated. More Complicated Some pictures will help us understand limits of sums, Functions differences, quotients, and products. The Derivative of a Sum is the Sum of the Derivatives

MAT 4810: Calculus for Elementary and Middle School Teachers

Computing Derivatives & Integrals Theorem Part I

Charles If f and g are differentiable at x, then so is f + g, and Delman [f + g]0(x) = f 0(x) + g 0(x) = [f 0 + g 0](x).

The Definition & Meaning of the Derivative This one is simple! Function

Local Coordinates

Computing Derivatives of More Complicated Functions Proof that [f + g]0 = f 0 + g 0

MAT 4810: Calculus for Elementary and Middle School y z Teachers y D z D } } Computing } } Derivatives & } } Integrals Part I f(x) g(x) Charles } } Delman f(x+ x) g(x+ x) The Definition D D & Meaning of the Derivative Function

Local Coordinates Proof. Computing 0 ∆[y+z] ∆y+∆z Derivatives of [f + g] (x) = lim∆x→0 = lim∆x→0 = More ∆x ∆x Complicated ∆y Functions ∆z 0 0 lim∆x→0 ∆x + lim∆x→0 ∆x = f (x) + g (x) The Derivative of a Product is Not the Product of the Derivatives

MAT 4810: Calculus for Elementary and Middle Theorem School Teachers If f and g are differentiable at x, then so is fg, and 0 0 0 0 0 Computing [fg] (x) = f (x)g (x) + f (x)g(x) = [fg + f g](x). Derivatives & Integrals Part I Corollary Charles Delman 0 0 If c ∈ R and g is differentiable at x, then [cg] (x) = cg (x). The Definition & Meaning of the Derivative Proof of corollary. Function Local The derivative of a constant function is the constant function Coordinates with output 0: if f (x) = c, then f 0(x) = 0. Thus, applying the Computing Derivatives of product rule, we obtain More Complicated [cg]0(x) = cg 0(x) + 0 · g(x) = cg 0(x) Functions Proof that [fg]0 = fg 0 + f 0g

MAT 4810: Calculus for Elementary and Middle } School z y z . } D D Dy Dz Teachers }

Computing Derivatives & g(x+Dx) g(x) z yz Dy .z Integrals Part I y Dy } }

} } Charles Delman

} f(x) f(x+Dx) The Definition & Meaning of the Derivative Function

Local Proof. Coordinates [fg]0(x) = lim ∆[yz] = lim y∆z+∆y·z+∆y·∆z = Computing ∆x→0 ∆x ∆x→0 ∆x Derivatives of More ∆z ∆y ∆y Complicated y · lim∆x→0 ∆x + lim∆x→0 ∆x · z + lim∆x→0 ∆x · lim∆x→0 ∆z = Functions f (x)g 0(x) + f 0(x)g(x) + f 0(x) · 0 = [fg 0 + f 0g](x) Examples

MAT 4810: Calculus for Elementary 4 0 3 and Middle Let h(x) = x . We will verify that h (x) = 4x . School Teachers Consider f (x) = x 3 and g(x) = x. h(x) = f (x)g(x). Computing From the definition of the derivative, we have computed Derivatives & 0 2 0 Integrals that f (x) = 3x and g (x) = 1. Part I Therefore, by the product rule, Charles 0 0 0 3 2 3 Delman h (x) = f (x)g (x) + f (x)g(x) = x · 1 + 3x · x = 4x . 4 0 The Definition Let f (x) = 5x . Compute f (x)? & Meaning of the Derivative Let g(x) = x 4. Then f = 5g. Function By the special case of the product rule in which one Local 0 0 3 3 Coordinates function is constant, f (x) = 5g (x) = 5(4x ) = 20x . Computing 4 3 0 Derivatives of Let f (x) = 5x + x + 2. Compute f (x). More Complicated By the sum rule, f 0(x) = 20x 3 + 3x 2 + 0 = 20x 3 + 3x 2. Functions