STROUD Worked examples and exercises are in the text PROGRAMME F11 DIFFERENTIATION.

Slides:



Advertisements
Similar presentations
Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve. 2.3 Derivatives of Trigonometric.
Advertisements

Copyright © Cengage Learning. All rights reserved. Differentiation 2.
The derivative and the tangent line problem (2.1) October 8th, 2012.
DIFFERENTIATION & INTEGRATION CHAPTER 4.  Differentiation is the process of finding the derivative of a function.  Derivative of INTRODUCTION TO DIFFERENTIATION.
Section 3.5a. Evaluate the limits: Graphically Graphin What happens when you trace close to x = 0? Tabular Support Use TblStart = 0, Tbl = 0.01 What does.
Copyright © Cengage Learning. All rights reserved. 3 Differentiation Rules.
The Power Rule  If we are given a power function:  Then, we can find its derivative using the following shortcut rule, called the POWER RULE:
Dr. Caulk FPFSC 145 University Calculus: Early Transcendentals, Second Edition By Hass, Weir, Thomas.
STROUD Worked examples and exercises are in the text PROGRAMME F11 DIFFERENTIATION = slope finding.
DIFFERENTIATION Differentiation is about rates of change. Differentiation is all about finding rates of change of one quantity compared to another. We.
3.5 – Derivative of Trigonometric Functions
1 The Product and Quotient Rules and Higher Order Derivatives Section 2.3.
STROUD Worked examples and exercises are in the text PROGRAMME 14 SERIES 2.
Tangents and Normals The equation of a tangent and normal takes the form of a straight line i.e. To find the equation you need to find a value for x, y.
DIFFERENTIATION RULES
Derivatives of Parametric Equations
Section 3.1 Derivatives of Polynomials and Exponential Functions  Goals Learn formulas for the derivatives ofLearn formulas for the derivatives of  Constant.
The Derivative. Definition Example (1) Find the derivative of f(x) = 4 at any point x.
STROUD Worked examples and exercises are in the text PROGRAMME F11 DIFFERENTIATION.
2-1 The Derivative and the Tangent Line Problem 2-2 Basic Differentiation Rules and Rates of Change 2-3 Product/Quotient Rule and Higher-Order Derivatives.
3.3 Rules for Differentiation Colorado National Monument.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Product and Quotient Rules for Differentiation.
STROUD Worked examples and exercises are in the text PROGRAMME 8 DIFFERENTIATION APPLICATIONS 1.
Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.
Some needed trig identities: Trig Derivatives Graph y 1 = sin x and y 2 = nderiv (sin x) What do you notice?
Sec. 3.3: Rules of Differentiation. The following rules allow you to find derivatives without the direct use of the limit definition. The Constant Rule.
3 DIFFERENTIATION RULES. We have:  Seen how to interpret derivatives as slopes and rates of change  Seen how to estimate derivatives of functions given.
Understanding the difference between an engineer and a scientist There are many similarities and differences.
STROUD Worked examples and exercises are in the text Programme 9: Tangents, normals and curvature TANGENTS, NORMALS AND CURVATURE PROGRAMME 9.
Math 1411 Chapter 2: Differentiation 2.3 Product and Quotient Rules and Higher-Order Derivatives.
2.1 The Derivative and the Tangent Line Problem.
STROUD Worked examples and exercises are in the text Programme 3: Hyperbolic functions HYPERBOLIC FUNCTIONS PROGRAMME 3.
STROUD Worked examples and exercises are in the text Programme F12: Differentiation PROGRAMME F12 DIFFERENTIATION.
STROUD Worked examples and exercises are in the text PROGRAMME 3 HYPERBOLIC FUNCTIONS.
2.3 Basic Differentiation Formulas
AP Calculus 3.2 Basic Differentiation Rules Objective: Know and apply the basic rules of differentiation Constant Rule Power Rule Sum and Difference Rule.
§ 4.2 The Exponential Function e x.
Algebra and Functions.
Homework, Page 460 Prove the algebraic identity. 1.
London Bridge, Lake Havasu City, Arizona
Copyright © Cengage Learning. All rights reserved.
3.1 Polynomial & Exponential Derivatives
2.3 Basic Differentiation Formulas
Calculus I (MAT 145) Dr. Day Friday September 29, 2017
Differentiating Polynomials & Equations of Tangents & Normals
DIFFERENTIATION APPLICATIONS 1
47 – Derivatives of Trigonometric Functions No Calculator
The Gradient at a point on a Curve
PROGRAMME F11 DIFFERENTIATION.
Differentiation with Trig – Outcomes
2.2 Rules for Differentiation
Chapter 3 Derivatives.
3.2: Rules for Differentiation
Copyright © Cengage Learning. All rights reserved.
London Bridge, Lake Havasu City, Arizona
2.4 The Chain Rule.
Differentiation Rules
Exam2: Differentiation
Copyright © Cengage Learning. All rights reserved.
DO NOW 14.6: Sum and Difference Formulas (PC 5.4)
2.1 The Derivative and the Slope of a Graph
Find the derivative Find the derivative at the following point.
Exam2: Differentiation
Copyright © Cengage Learning. All rights reserved.
difference in the y-values
Gradients and Tangents
Sum and Difference Formulas (Section 5-4)
2.5 Basic Differentiation Properties
London Bridge, Lake Havasu City, Arizona
Presentation transcript:

STROUD Worked examples and exercises are in the text PROGRAMME F11 DIFFERENTIATION

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method [optional] Programme F11: Differentiation

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph Programme F11: Differentiation The gradient of the sloping line straight line in the figure is defined as: the vertical distance the line rises and falls between the two points P and Q the horizontal distance between P and Q

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph Programme F11: Differentiation The gradient of the sloping straight line in the figure is given as:

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Second derivatives Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text The gradient of a curve at a given point Programme F11: Differentiation The gradient of a curve between two points will depend on the points chosen:

STROUD Worked examples and exercises are in the text The gradient of a curve at a given point The gradient of a curve at a point P is defined to be the gradient of the tangent at that point:

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function (Second derivatives –MOVED to a later set of slides) Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve Programme F11: Differentiation The gradient of the chord PQ is and the gradient of the tangent at P is

STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve Programme F11: Differentiation As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines Two curves Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines Programme F11: Differentiation (a)

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two straight lines (b) Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two curves Programme F11: Differentiation (a) so

STROUD Worked examples and exercises are in the text Derivatives of powers of x Two curves Programme F11: Differentiation (b) so

STROUD Worked examples and exercises are in the text Derivatives of powers of x A clear pattern is emerging:

STROUD Worked examples and exercises are in the text Algebraic determination of the gradient of a curve At Q: So As Therefore called the derivative of y with respect to x.

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Differentiation of polynomials Programme F11: Differentiation To differentiate a polynomial, we differentiate each term in turn:

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Derivatives – an alternative notation Programme F11: Differentiation The double statement: can be written as:

STROUD Worked examples and exercises are in the text Towards derivatives of trigonometric functions (JAB) Limiting value of is 1 I showed this in an earlier lecture by a rough argument. Following slide includes most of a rigorous argument. Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Programme F11: Differentiation Area of triangle POA is: Area of sector POA is: Area of triangle POT is: Therefore: That is ((using fact that the cosine tends to 1 -- JAB)):

STROUD Worked examples and exercises are in the text Derivatives of trigonometric functions and … Programme F11: Differentiation The table of standard derivatives can be extended to include trigonometric and the exponential functions: ((JAB:)) The trig cases use the identities for finding sine and cosine of the sum of two angles, and an approximation I gave earlier for the cosine of a small angle. (Shown in class).

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Differentiation of products of functions Programme F11: Differentiation Given the product of functions of x: then: This is called the product rule.

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Differentiation of a quotient of two functions Programme F11: Differentiation Given the quotient of functions of x: then: This is called the quotient rule.

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Functions of a function Differentiation of a function of a function To differentiate a function of a function we employ the chain rule. If y is a function of u which is itself a function of x so that: Then: This is called the chain rule. Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Functions of a function Differentiation of a function of a function Programme F11: Differentiation Many functions of a function can be differentiated at sight by a slight modification to the list of standard derivatives:

STROUD Worked examples and exercises are in the text The gradient of a straight-line graph The gradient of a curve at a given point Algebraic determination of the gradient of a curve Derivatives of powers of x Differentiation of polynomials Derivatives – an alternative notation Differentiation of products of functions Differentiation of a quotient of two functions Functions of a function Newton-Raphson iterative method [optional] Programme F11: Differentiation

STROUD Worked examples and exercises are in the text Newton-Raphson iterative method [OPTIONAL] Tabular display of results Programme F11: Differentiation Given that x 0 is an approximate solution to the equation f(x) = 0 then a better solution is given as x 1, where: This gives rise to a series of improving solutions by iteration using: A tabular display of improving solutions can be produced in a spreadsheet.