DOCSLIB.ORG

17 Stationary Points of Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
17 Stationary Points of Functions - 17 Stationary Points of Functions A stationary point of a curve is a point on the curve at which the gradient is zero. In this section we will focus on the definitions of different types of stationary points of functions and how to find them. By the end of this section, you should have the following skills: • An understanding of the definition of types of stationary points, in- cluding: local maxima, local minima and points of inflection. • Find stationary points of a function using differentiation, including: local maxima, local minima and points of inflection. • Test points to see if they are a stationary point. • Test stationary points to determine their type of stationary point, using first and second derivatives. 17.1 Local Maximum and Local Minimum Let f(x) be a function. It is useful to be able to find and classify local maxima and minima. These are points at which the function switches from increasing to decreasing and vice-versa. 17.1.1 Local Minimum Here the graph is decreasing and then increasing as x increases through the local minimum. The point x = b in the diagram is a local minimum. 17.1.2 Local Maximum In contrast, the graph is increasing and then decreasing as x increases through the local maximum. The point x = a in the diagram is a local maximum. 1 Local minimum and local maximum. Note that such a point as a is only a local maximum as it is possible that the function takes values elsewhere which are greater than the value at a. In our example, there are clearly values for the function which are greater than the value at x = a. Similarly for a local minimum. Example 1 1. f(x) = 1 has a local maximum and a local minimum at every point! 2. Any linear function f(x) = ax+b; a 6= 0; b constants has neither local maxima nor local minima. 3. f(x) = x2 has a local minimum at x = 0. 4. f(x) = −x2 has a local maximum at x = 0. 5. a quadratic has either one local minimum or one local maximum (but not both). 6. f(x) = x3 − 3x has a local maximum at x = −1 and a local mini- mum at x = 1. 2 7. f(x) = sin(x) has local maxima at x = π=2; 5π=2 9π=2;::: and has local minima at x = 3π=2; 7π=2 11π=2;:::. 8. f(x) = tan(x) has no local maxima or minima. 9. f(x) = ex has no local maxima or minima. 10. f(x) = ln(x) has no local maxima or minima. 11. f(x) = 1=x has no local maxima or minima. 12. f(x) = ex − x has a local minimum at x = 0 where f(0) = 1. Note that this shows that ex − x ≥ 1 ) ex ≥ 1 + x for all x. 3 13. 1 f(x) = (x − 1)(x − 2) has a local maximum at x = 3=2 where f(3=2) = −4. Note carefully that a local maximum is not a global maximum in general. For example f(x) = x3 −3x has a local maximum of f(−1) = 2 at x = −1 but this is not a maximum for the function as clearly there are points at which this function is greater than 2. What is important here is that all other points x sufficiently near to −1 have f(x) ≤ 2. Similarly for local minima, 4 x = 1 is a local minimum for f(x) = x3 − 3x but not a global minimum. Exercise 1 Each of the following functions f(x) has a local maximum or mini- mum as indicated. Which of these is a global maximum or minimum? (a) f(x) = x2 has a local minimum at x = 0. (b) f(x) = −x2 has a local maximum at x = 0. (c) f(x) = sin(x) has a local maximum at x = π=2. (d) f(x) = sin(x) has a local minimum at x = 3π=2. (e) f(x) = 1=(x − 1)(x − 2) has a local maximum at x = 3=2. (f) f(x) = ex − x has a local minimum at x = 0. Solutions to exercise 1 (a) f(x) = x2 has a local minimum at x = 0. (b) f(x) = −x2 has a local maximum at x = 0. 5 Graph of x2 and −x2. Clearly y = x2 has a global maximum at x = 0 whereas y = −x2 has a local minimum at x = 0. (c) f(x) = sin(x) has a local maximum at x = π=2. (d) f(x) = sin(x) has a local minimum at x = 3π=2. Since sin(π=2) = 1 and sin(3π=2) = −1 and for all x, −1 ≤ sin(x) ≤ 1 we see that x = π=2 is a global maximum and x = 3π=2 is a global minimum. (e) f(x) = 1=((x − 1)(x − 2)) has a local maximum at x = 3=2. The vertical lines at x = 1 and x = 2 are asymptotes. 6 We see that x = 3=2 is not a global maximum as for example f(0) = 1=2 > f(3=2): (f) f(x) = ex − x has a local minimum at x = 0. Note that if y = ex−x then dy=dx = ex−1 and so for x > 0; dy=dx > 0 and for x < 0; dy=dx < 0, hence the function is decreasing for x < 0 and increasing for x > 0, so x = 0 must be a global minimum. 7 17.2 Finding Local Maxima and Minima 17.2.1 Reminder about increasing and decreasing functions The following results from Increasing and Decreasing Functions are used in this section. Note that these results are true if f(x) is differentiable and f 0(x) is continuous, which we assume. 1. If f(x) is increasing for a range of values of x then df ≥ 0 dx over that range of values of x. 2. If f(x) is decreasing for a range of values of x then df ≤ 0 dx over that range of values of x. 17.2.2 Local Maxima We deal first with local maxima - the details for local minima are similar. First recall that a local maximum at x = a occurs when a function changes from increasing to decreasing i.e. • in an interval before b ≤ x ≤ a, f(x) is increasing ) df=dx ≥ 0; b ≤ x ≤ a and • in an interval after a ≤ x ≤ c, f(x) is decreasing ) df=dx ≤ 0; a ≤ x ≤ c. 8 Local Maximum. Since df=dx ≥ 0; df=dx ≤ 0; x = a we must have: If x = a is a local maximum then df = 0; x = a dx and df=dx decreases from positive to negative as x goes through a. 17.2.3 Local Minima Local Minimum. In a similar way we can show that if x = a is a local minimum then df = 0; x = a dx and df=dx increases from negative to positive as x goes through a. 17.2.4 Summary We have shown that if x = a is a local maximum or a local minimum then df = 0; x = a: dx This tells us that the tangent at x = a is horizontal i.e. parallel to the x-axis. 9 17.3 Stationary Points Any point x = a such that df = 0; x = a dx is called a stationary point. This is because the rate of change at x = a is 0 i.e. instantaneously stationary. So a local maximum or minimum is a stationary point. Note that for a local maximum or a local minimum the sign of df=dx changes as x goes through the stationary point. 17.3.1 Points of inflection There is another type of stationary point called a point of inflection. At these points we still have df=dx = 0 but df=dx does not change sign as x goes through a. For example y = x3 has a stationary point at x = 0, but dy=dx = 3x2 ≥ 0 for all x. So dy=dx does not change sign. Similarly, y = −x3 has a point of inflection at x = 0 as dy=dx ≤ 0 for all x. Figure 1: y = x3. 10 17.4 Finding Stationary Points If f(x) is a function we solve the equation df = 0 dx for x. The values of x we find are the stationary points of f(x). Example 2 Find the stationary points for the following functions f(x): (a) f(x) = x2 − 5x + 6. (b) f(x) = x3 − 6x2 + 7. (c) f(x) = x3 + 3x + 5. (d) f(x) = sin(x). (e) f(x) = ex − 2x. (f) f(x) = (x − 1)=(x2 + 5x + 3). (g) f(x) = ln(x3 − 6x2 − 15x + 1). (h) f(x) = 3x5 − 5x3 + 4. p (i) f(x) = 7 + (2x2 − 10x) x. Solution. (a) df = 2x − 5: dx The stationary points are given by x such that df 5 = 0 ) x = : dx 2 So one stationary point x = 5=2. 11 Graph of x2 − 5x + 6. (b) df = 3x2 − 12x: dx The stationary points are given by x such that df = 0 ) 3x2 − 12x = 3x(x − 4) = 0 ) x = 0; x = 4: dx So the stationary points are x = 0; 4. Graph of x3 − 6x2 + 7. 12 (c) df = 3x2 + 3: dx The stationary points are given by x such that df = 0 ) 3x2 + 3 = 0 ) x2 = −1: dx But there are no real number solutions of this equation. So no sta- tionary points. (d) df = cos(x): dx The stationary points are given by x such that df = 0 ) cos(x) = 0 ) x = ±π=2; ±3π=2; ±5π=2;:::: dx There are an infinite number of stationary points.
Load more

Recommended publications
  • Tutorial 8: Solutions
    Tutorial 8: Solutions Applications of the Derivative 1. We are asked to find the absolute maximum and minimum values of f on the given interval, and state where those values occur: (a) f(x) = 2x3 + 3x2 12x on [1; 4]. − The absolute extrema occur either at a critical point (stationary point or point of non-differentiability) or at the endpoints. The stationary points are given by f 0(x) = 0, which in this case gives 6x2 + 6x 12 = 0 = x = 1; x = 2: − ) − Checking the values at the critical points (only x = 1 is in the interval [1; 4]) and endpoints: f(1) = 7; f(4) = 128: − Therefore the absolute maximum occurs at x = 4 and is given by f(4) = 124 and the absolute minimum occurs at x = 1 and is given by f(1) = 7. − (b) f(x) = (x2 + x)2=3 on [ 2; 3] − As before, we find the critical points. The derivative is given by 2 2x + 1 f 0(x) = 3 (x2 + x)1=2 and hence we have a stationary point when 2x + 1 = 0 and points of non- differentiability whenever x2 + x = 0. Solving these, we get the three critical points, x = 1=2; and x = 0; 1: − − Checking the value of the function at these critical points and the endpoints: f( 2) = 22=3; f( 1) = 0; f( 1=2) = 2−4=3; f(0) = 0; f(3) = (12)2=3: − − − Hence the absolute minimum occurs at either x = 1 or x = 0 since in both − these cases the minimum value is 0, while the absolute maximum occurs at x = 3 and is f(3) = (12)2=3.
    [Show full text]
  • Lecture Notes – 1
    Optimization Methods: Optimization using Calculus-Stationary Points 1 Module - 2 Lecture Notes – 1 Stationary points: Functions of Single and Two Variables Introduction In this session, stationary points of a function are defined. The necessary and sufficient conditions for the relative maximum of a function of single or two variables are also discussed. The global optimum is also defined in comparison to the relative or local optimum. Stationary points For a continuous and differentiable function f(x) a stationary point x* is a point at which the slope of the function vanishes, i.e. f ’(x) = 0 at x = x*, where x* belongs to its domain of definition. minimum maximum inflection point Fig. 1 A stationary point may be a minimum, maximum or an inflection point (Fig. 1). Relative and Global Optimum A function is said to have a relative or local minimum at x = x* if f ()xfxh**≤+ ( )for all sufficiently small positive and negative values of h, i.e. in the near vicinity of the point x*. Similarly a point x* is called a relative or local maximum if f ()xfxh**≥+ ( )for all values of h sufficiently close to zero. A function is said to have a global or absolute minimum at x = x* if f ()xfx* ≤ ()for all x in the domain over which f(x) is defined. Similarly, a function is D Nagesh Kumar, IISc, Bangalore M2L1 Optimization Methods: Optimization using Calculus-Stationary Points 2 said to have a global or absolute maximum at x = x* if f ()xfx* ≥ ()for all x in the domain over which f(x) is defined.
    [Show full text]
  • The Theory of Stationary Point Processes By
    THE THEORY OF STATIONARY POINT PROCESSES BY FREDERICK J. BEUTLER and OSCAR A. Z. LENEMAN (1) The University of Michigan, Ann Arbor, Michigan, U.S.A. (3) Table of contents Abstract .................................... 159 1.0. Introduction and Summary ......................... 160 2.0. Stationarity properties for point processes ................... 161 2.1. Forward recurrence times and points in an interval ............. 163 2.2. Backward recurrence times and stationarity ................ 167 2.3. Equivalent stationarity conditions .................... 170 3.0. Distribution functions, moments, and sample averages of the s.p.p ......... 172 3.1. Convexity and absolute continuity .................... 172 3.2. Existence and global properties of moments ................ 174 3.3. First and second moments ........................ 176 3.4. Interval statistics and the computation of moments ............ 178 3.5. Distribution of the t n .......................... 182 3.6. An ergodic theorem ........................... 184 4.0. Classes and examples of stationary point processes ............... 185 4.1. Poisson processes ............................ 186 4.2. Periodic processes ........................... 187 4.3. Compound processes .......................... 189 4.4. The generalized skip process ....................... 189 4.5. Jitte processes ............................ 192 4.6. ~deprendent identically distributed intervals ................ 193 Acknowledgments ............................... 196 References ................................... 196 Abstract An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Q) Presently at Massachusetts Institute of Technology, Lexington, Mass., U.S.A. (2) This work was supported by the National Aeronautics and Space Administration under research grant NsG-2-59. 11 - 662901. Aeta mathematica. 116. Imprlm$ le 19 septembre 1966.
    [Show full text]
  • The First Derivative and Stationary Points
    Mathematics Learning Centre The first derivative and stationary points Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 The first derivative and stationary points dy The derivative of a function y = f(x) tell us a lot about the shape of a curve. In this dx section we will discuss the concepts of stationary points and increasing and decreasing functions. However, we will limit our discussion to functions y = f(x) which are well behaved. Certain functions cause technical difficulties so we will concentrate on those that don’t! The first derivative dy The derivative, dx ,isthe slope of the tangent to the curve y = f(x)atthe point x.Ifwe know about the derivative we can deduce a lot about the curve itself. Increasing functions dy If > 0 for all values of x in an interval I, then we know that the slope of the tangent dx to the curve is positive for all values of x in I and so the function y = f(x)isincreasing on the interval I. 3 For example, let y = x + x, then y dy 1 =3x2 +1> 0 for all values of x. dx That is, the slope of the tangent to the x curve is positive for all values of x. So, –1 0 1 y = x3 + x is an increasing function for all values of x. –1 The graph of y = x3 + x. We know that the function y = x2 is increasing for x>0. We can work this out from the derivative. 2 If y = x then y dy 2 =2x>0 for all x>0.
    [Show full text]
  • EE2 Maths: Stationary Points
    EE2 Maths: Stationary Points df Univariate case: When we maximize f(x) (solving for the x0 such that dx = 0) the d df gradient is zero at these points. What about the rate of change of gradient: dx ( dx ) at the minimum x0? For a minimum the gradient increases as x0 ! x0 + ∆x (∆x > 0). It follows d2f d2f that dx2 > 0. The opposite is true for a maximum: dx2 < 0, the gradient decreases upon d2f positive steps away from x0. For a point of inflection dx2 = 0. r @f @f Multivariate case: Stationary points occur when f = 0. In 2-d this is ( @x ; @y ) = 0, @f @f namely, a generalization of the univariate case. Recall that df = @x dx + @y dy can be written as df = ds · rf where ds = (dx; dy). If rf = 0 at (x0; y0) then any infinitesimal step ds away from (x0; y0) will still leave f unchanged, i.e. df = 0. There are three types of stationary points of f(x; y): Maxima, Minima and Saddle Points. We'll draw some of their properties on the board. We will now attempt to find ways of identifying the character of each of the stationary points of f(x; y). Consider a Taylor expansion about a stationary point (x0; y0). We know that rf = 0 at (x0; y0) so writing (∆x; ∆y) = (x − x0; y − y0) we find: − ∆f = f(x; y)[ f(x0; y0) ] 1 @2f @2f @2f ' 0 + (∆x)2 + 2 ∆x∆y + (∆y)2 : (1) 2! @x2 @x@y @y2 Maxima: At a maximum all small steps away from (x0; y0) lead to ∆f < 0.
    [Show full text]
  • Applications of Differentiation Applications of Differentiation– a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 12 Applications of differentiation Applications of differentiation – A guide for teachers (Years 11–12) Principal author: Dr Michael Evans, AMSI Peter Brown, University of NSW Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Graph sketching ....................................... 5 More graph sketching ..................................
    [Show full text]
  • Lecture 3: 20 September 2018 3.1 Taylor Series Approximation
    COS 597G: Toward Theoretical Understanding of Deep Learning Fall 2018 Lecture 3: 20 September 2018 Lecturer: Sanjeev Arora Scribe: Abhishek Bhrushundi, Pritish Sahu, Wenjia Zhang Recall the basic form of an optimization problem: min (or max) f(x) subject to x 2 B; where B ⊆ Rd and f : Rd ! R is the objective function1. Gradient descent (ascent) and its variants are a popular choice for finding approximate solutions to these sort of problems. In fact, it turns out that these methods also work well empirically even when the objective function f is non-convex, most notably when training neural networks with various loss functions. In this lecture we will explore some of the basics ideas behind gradient descent, try to understand its limitations, and also discuss some of its popular variants that try to get around these limitations. We begin with the Taylor series approximation of functions which serves as a starting point for these methods. 3.1 Taylor series approximation We begin by recalling the Taylor series for univariate real-valued functions from Calculus 101: if f : R ! R is infinitely differentiable at x 2 R then the Taylor series for f at x is the following power series (∆x)2 (∆x)k f(x) + f 0(x)∆x + f 00(x) + ::: + f (k)(x) + ::: 2! k! Truncating this power series at some power (∆x)k results in a polynomial that approximates f around the point x. In particular, for small ∆x, (∆x)2 (∆x)k f(x + ∆x) ≈ f(x) + f 0(x)∆x + f 00(x) + ::: + f (k)(x) 2! k! Here the error of the approximation goes to zero at least as fast as (∆x)k as ∆x ! 0.
    [Show full text]
  • Motion in a Straight Line Motion in a Straight Line – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 17 Motion in a straight line Motion in a straight line – A guide for teachers (Years 11–12) Principal author: Dr Michael Evans AMSI Peter Brown, University of NSW Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 6 Position, displacement and distance ........................... 6 Constant velocity .....................................
    [Show full text]
  • 1 Overview 2 a Characterization of Convex Functions
    AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 — February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the simplex method for finding optimal solutions. In this lecture we return to convex optimization and give a characterization of optimality. The main theorem we will prove today shows that for a convex function f : S ! R defined over a convex set S, a stationary point (the point x¯ for which rf(x¯)) is a global minimum. We we will review basic definitions and properties from multivariate calculus, prove some important properties of convex functions, and obtain various characterizations of optimality. 2 A Characterization of Convex Functions Recall the definitions we introduced in the second lecture of convex sets and convex functions: Definition. A set S is called a convex set if any two points in S contain their line, i.e. for any x1; x2 2 S we have that λx1 + (1 − λ)x2 2 S for any λ 2 [0; 1]. n Definition. For a convex set S ⊆ R , we say that a function f : S ! R is convex on S if for any two points x1; x2 2 S and any λ 2 [0; 1] we have that: f (λx1 + (1 − λ)x2) ≤ λf(x1) + 1 − λ f(x2): We will first give an important characterization of convex function. To so, we need to characterize multivariate functions via their Taylor expansion. 2.1 First-order Taylor approximation n n Definition. The gradient of a function f : R ! R at point x 2 R denoted rf(x¯) is: @f(x) @f(x)| rf(x) := ;:::; @x1 @xn First-order Taylor expansion.
    [Show full text]
  • Implicit Differentiation
    Learning Development Service Differentiation 2 To book an appointment email: [email protected] Who is this workshop for? • Scientists-physicists, chemists, social scientists etc. • Engineers • Economists • Calculus is used almost everywhere! Learning Development Service What we covered last time : • What is differentiation? Differentiation from first principles. • Differentiating simple functions • Product and quotient rules • Chain rule • Using differentiation tables • Finding the maxima and minima of functions Learning Development Service What we will cover this time: • Second Order Differentiation • Maxima and Minima (Revisited) • Parametric Differentiation • Implicit Differentiation Learning Development Service The Chain Rule by Example (i) Differentiate : Set: The differential is then: Learning Development Service The Chain Rule: Examples Differentiate: 2 1/3 (i) y x 3x 1 (ii) y sincos x (iii) y cos3x 1 3 2x 1 (iv) y 2x 1 Learning Development Service Revisit: maxima and minima Learning Development Service Finding the maxima and minima Learning Development Service The second derivative • The second derivative is found by taking the derivative of a function twice: dy d 2 y y f (x) f (x) f (x) dx dx2 • The second derivative can be used to classify a stationary point Learning Development Service The second derivative • If x0 is the position of a stationary point, the point is a maximum if the second derivative is <0 : d 2 y 0 dx2 and a minimum if: d 2 y 0 dx2 • The nature of the stationary point is inconclusive if the second derivative=0. Learning Development Service The second derivative: Examples 3 • Find the stationary points of the function f (x) x 6x and classify the stationary points.
    [Show full text]
  • Calculus Online Textbook Chapter 13 Sections 13.5 to 13.7
    13.5 The Chain Rule 34 A quadratic function ax2 + by2 + cx + dy has the gradi- 42 f=x x = cos 2t y = sin 2t ents shown in Figure B. Estimate a, b, c, d and sketch two level curves. 43 f=x2-y2 x=xo+2t y=yo+3t 35 The level curves of f(x, y) are circles around (1, 1). The 44 f=xy x=t2+1 y=3 curve f = c has radius 2c. What is f? What is grad f at (0, O)? 45 f=ln xyz x = e' y = e2' = e-' 36 Suppose grad f is tangent to the hyperbolas xy = constant 46 f=2x2+3y2+z2 x=t y=t2 Z=t3 in Figure C. Draw three level curves off(x, y). Is lgrad f 1 larger 47 (a) Find df/ds and df/dt for the roller-coaster f = xy along at P or Q? Is lgrad f 1 constant along the hyperbolas? Choose the path x = cos 2t, y = sin 2t. (b) Change to f = x2 + y2 and a function that could bef: x2 + y2, x2 - y2, xy, x2y2. explain why the slope is zero. 37 Repeat Problem 36, if grad f is perpendicular to the hyper- bolas in Figure C. 48 The distance D from (x, y) to (1, 2) has D2 = (x - + (y - 2)2. Show that aD/ax = (X- l)/D and dD/ay = 38 Iff = 0, 1, 2 at the points (0, I), (1, O), (2, I), estimate grad f (y - 2)/D and [grad Dl = 1.
    [Show full text]
  • Module C6 Describing Change – an Introduction to Differential Calculus 6 Table of Contents
    Module C6 – Describing change – an introduction to differential calculus Module C6 Describing change – an introduction to differential calculus 6 Table of Contents Introduction .................................................................................................................... 6.1 6.1 Rate of change – the problem of the curve ............................................................ 6.2 6.2 Instantaneous rates of change and the derivative function .................................... 6.15 6.3 Shortcuts for differentiation ................................................................................... 6.19 6.3.1 Polynomial and other power functions ............................................................. 6.19 6.3.2 Exponential functions ....................................................................................... 6.28 6.3.3 Logarithmic functions ...................................................................................... 6.30 6.3.4 Trigonometric functions ................................................................................... 6.34 6.3.5 Where can’t you find a derivative of a function? ............................................. 6.37 6.4 Some applications of differential calculus ............................................................. 6.40 6.4.1 Displacement-velocity-acceleration: when derivatives are meaningful in their own right ........................................................................................................... 6.41 6.4.2 Twists and turns ...............................................................................................
    [Show full text]