Advanced Topics in Physics: Velocity, Speed, and Rates of Change. Photo credit: Dr. Persin, Founder of Lnk2Lrn.

Advanced Topics in Physics - Calculus Application. The Rate of Change of a Function: Consider a function x = f(t). t x A ( t, f(t) ) ( t +  t, f(t +  t) ) B + t t +  t f(t +  t) f(t)

The Slope of the Secant Line AB A ( t, f(t) ) ( t +  t, f(t +  t) ) B + t t +  t f(t +  t) ft)  y f(t+  t) - f(t) m = =  x (t+  t) - t Gives us the average rate of change of position versus time, or the average velocity, v avg, of an object. More useful is instantaneous rate of change. Position x t Time

1630’s: Descartes and Fermat discover the general rule for the slope of tangent to a polynomial using the Limit as  t  0. René Descartes “I think, therefore I am.” Pierre de Fermat x n + y n = z n has no non-zero integer solutions for x, y and z when n > 2.

Instantaneous Rate of Change This is the Slope of the Tangent to the Curve, given by the Limit as  t  0, or f(t+  t) - f(t) (t+  t) - t Lim  t  0 dxdt = Also known as the first derivative of the function with respect to t. Or, the rate of change of the function based on slight changes in t. This the instantaneous velocity, v. = v

Rules of Differentiation Constant Rule: If f(x) = k, then f ΄ (x) = 0. e.g. Suppose f(x) = 3. What is f ΄ (x)? Power Function Rule. If f(x) = cx n, then f ΄ (x) = cnx n-1 e.g. Suppose f(x) = 3x 2, what is f ΄ (x)? Sum-Difference Rule. If f(x) = g(x) ± h(x), then f ΄ (x) = g ΄ (x) ± h ΄ (x) e.g. Suppose f(x) = x 2 + 3x 3, what is f ΄ (x)? e.g. Suppose f(x) = 17 – 4x, what is f ΄ (x)?

More rules of Differentiation Product Rule: If f(x) = g(x)h(x), then f ΄ (x) = g(x)h’(x) + h(x)g’(x) e.g. Suppose f(x) = (4x 3 )(5-x 2 ). What is f’(x)? Quotient Rule: If f(x) = g(x) / h(x), then f’(x) = [h(x)g’(x) -g(x)h’(x) ] / h(x) 2 e.g. Suppose f(x) = 2x 2 / (x-2). What is f’(x)?

More rules of differentiation Log Rule: If f(x) = ln( g(x) ), then f’(x) = g’(x) / g(x) e.g. Suppose f(x) = ln(x). What is f’(x)? Exponential-Function Rule: If f(x) = e g(x), then f’(x) = g’(x)e g(x) e.g. Suppose f(x) = e 3x, what is f’(x)?

Integration The derivative stemmed from the need to compute the slope of a function f(x). Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis. For example, suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10. Then, a numerical solution for this integral obviously exists, equals to 20. x f(x) 010 2

Integration cont. The integral of f(x) is defined as F(x) =  f(x) dx. F’(x) represents the “anti-derivative” of the function f(x). In other words, “F prime of x equals f of x.” F’(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668) John WallisJames Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670) Isaac Barrow

Derivative and the Integral first discovered by Newton (1666, unpublished); and later supported by Leibniz (1673) Isaac NewtonGottfried Leibniz

Joseph Fourier (1807): Put the emphasis on definite integrals (he invented the notation ) and defined them in terms of area between graph and x-axis.

A.L. Cauchy: First to define the integral as the limit of the summation Also the first (1823) to explicitly state and prove the second part of the FTC:

Bernhard Riemann ( ) On the representation of a function as a limit of a series. (Riemann Sum) Defined as limit of

The Fundamental Theorem of Calculus: If then S. F. LaCroix ( ):“Integral calculus is the inverse of differential calculus. Its goal is to restore the functions from their differential coefficients.” “When students find themselves stopped on a proof, the professor should restrain from immediately pointing out the solution. Let the students find it out for themselves; and the error corrected may be more profitable than several theorems proved.”

The Fundamental Theorem of Calculus: If then Vito Volterra, in 1881, found a function f with an anti-derivative F so that F΄(x) = f(x) for all x, but there is no interval over which the definite integral of f(x) exists.

Henri Lebesgue, in 1901, came up with a totally different way of defining integrals, using a “Step Function”, that is the same as the Riemann integral for well-behaved functions.

Rules of Integration Rule 1)  a dx = ax + c e.g. What is  2 dx = ? ______________ Rule 2)  x n dx = x n+1 / (n+1) + c e.g. What is  x 3 dx ? ________________ Rule 3)  a f(x) dx = a  f(x) dx e.g. What is  17 x 3 dx ? ________________ Rule 4) If u and v are functions of x,  (u+v) dx =  u dx +  v dx e.g. What is  (5 x x) dx ? ____________________________ Note that for each of these rules, we must add a constant of integration. To find the area under a curve, we use a Definite Integral. Find the area under the graph of f(x) = 7 - x 2 from x= -1 to x = 2

Basic Properties of Integrals Through this section we assume that all functions are continuous on a closed interval I = [a,b]. Below r is a real number, f and g are functions These properties of integrals follow from the definition of integrals as limits of Riemann sums.

Evaluating the Definite Integral Ex. Calculate

Substitution for Definite Integrals Ex. Calculate Notice limits change

Computing Area Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x 3 is nonnegative on [0, 2]. AntiderivativeFund. Thm. of Calculus

Derivatives of Functions of Higher Degree Ex 1: Suppose f(x) = (2x + 5) 3. Find f ’(x) and f ’’(x) after expanding the binomial. Solution: f(x) = (2x + 5) 3 =(2x) 3 + 3(2x) 2 ·5 + 3(2x) · = 8x x x f ’(x) = 24x x = 6(4x x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5) 2 f ’’(x) = 48x = 24(2x + 5) Notice similarities between the solutions and the original function?

For More Efficient Solutions of Problems of This Type We can use the Chain Rule If f (v) = v n and v is a function of x, then f (v) = nv n-1 dv. Now f(x) = (2x + 5) 3. Find f ’(x) and f ’’(x) using the Chain Rule

Lets Review the Following Before Going On Basic Rules of Differentiation The Product and Quotient Rules The Chain Rule Marginal Functions in Economics Higher-Order Derivatives Implicit Differentiation and Related Rates Differentials

Basic Differentiation Rules 1. Ex. 2. Ex.

Basic Differentiation Rules 3. Ex. 4. Ex.

More Differentiation Rules 5. Ex. Product Rule Derivative of the first function Derivative of the second function

More Differentiation Rules 6.Quotient Rule Sometimes remembered as:

More Differentiation Rules 6. Ex. Quotient Rule (cont.) Derivative of the numerator Derivative of the denominator

More Differentiation Rules 7. The Chain Rule Note: h(x) is a composite function. Another Version:

More Differentiation Rules The General Power Rule: Ex. The Chain Rule leads to

Chain Rule Example Ex.

Chain Rule Example Ex. Sub in for u

Higher Derivatives The second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative. DerivativeNotations Second Third Fourth nth

Example of Higher Derivatives Givenfind

Example of Higher Derivatives Givenfind

Implicit Differentiation y is expressed explicitly as a function of x. y is expressed implicitly as a function of x. To differentiate the implicit equation, we write f (x) in place of y to get:

Implicit Differentiation (cont.) Now differentiate using the chain rule: which can be written in the form subbing in y Solve for y’

Related Rates Look at how the rate of change of one quantity is related to the rate of change of another quantity. Ex.Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels west at 60 mi./hr. How fast is the distance between them changing after 2 hours? Note: The rate of change of the distance between them is related to the rate at which the cars are traveling.

Related Rates Steps to solve a related rate problem: 1. Assign a variable to each quantity. Draw a diagram if appropriate. 2. Write down the known values/rates. 3. Relate variables with an equation. 4. Differentiate the equation implicitly. 5. Plug in values and solve.

Ex. Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours? Distance = z x y From original relationship

Increments An increment in x represents a change from x 1 to x 2 and is defined by: Read “delta x” An increment in y represents a change in y and is defined by:

Differentials Let y = f (x) be a differentiable function, then the differential of x, denoted dx, is such that The differential of y, denoted dy, is Note:

Example Given