ECM Calculus and Geometry. Revision Notes

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1 ECM Calculus and Geometry Revision Notes Joshua Byrne Autumn 2011

2 Contents 1 The Real Numbers Notation Set Notation Open/Closed Intervals Coordinate Geometry Linear Equations Relationship between m and the Angle of Inclination Distance Between Two Points Distance Between a Point and a Line Conic Sections Expressing Curves in Cartesian and Parametric Form Quadratic Equations of the Conic Sections Finding Tangents and Normals to the Curves Directrices, Foci and Eccentricity Identifying a Conic Section by its Equation Functions Domains, Codomains and Ranges Algebra with Functions Composite Functions Odd and Even Functions Periodic Functions Monotonic Functions Trigonometric Functions Hyperbolic Functions Inverse Functions Piecewise Functions Sequences, Series and Limits Sequences Defining Sequences Convergence of Sequences Series and The Sigma Notation Series and Partial Sums Geometric Series Arithmetic Series Properties of Convergent Series Standard Series Limits Tests for Convergence and Divergence of a Series Power Series Limits and Continuity of Functions Definition of a Limit Left and Right Limits Differentiation Differentiation by First Principles Continuity and Differentiation Rules of Differentiation Implicit Differentiation Computing Higher Derivatives Parametric Differentiation Logarithmic Differentiation Differentiation of Inverse Functions Leibniz Rule for Repeated Differentiation of Products

3 8 Applications of Differentiation Linearization Extrema of Functions Convexity and Concavity Limits using Calculus - L Hôpital s Rule Intermediate Value Theorem Rolle s Theorem Mean Value Theorem Maclaurin Series Taylor Series Curve Sketching 18 These notes were compiled using Thomas Calculus, lecture notes uploaded to ELE, and internet resources.

4 1 The Real Numbers 1.1 Notation Natural Numbers 1, 2, 3... set N Integers...-2, -1, 0, 1, 2... set Z Rational Numbers Any numbers that can be written a set Q b Real Numbers Includes 2, π set R x is an element of the set R: x R Absolute Value: abs(x) = x 1.2 Set Notation The set of x in A is such that x satisfies some property B: {x A : x satisfied B} A is a subset of B: A B Intersection - in A and B: A B Union - in A or B: A B Complement - not in A: A : A = {x R : x / A} 1.3 Open/Closed Intervals Open interval: (a, b) = {x : a < x < b} Closed interval: [a, b] = {x : a x b} Half opened/closed interval: (a, b] = {x : a < x b} 2 Coordinate Geometry We can represent points in the plane as pairs (x, y) R 2 - this is very similar to the notation used in set theory. For example, {(x, y) R 2 : f(x, y) = 0} means the set of points that satisfy f(x, y) = Linear Equations The following are all versions of writing linear equations. {(x, y) : ax + by + c = 0} y = mx + c y y 1 = m(x x 1 ) ax + by + c = 0 is preferred. 2.2 Relationship between m and the Angle of Inclination m = increase in y = tan θ, where θ is the angle between the positive real axis and the line. increase in x Parallel lines have the same gradient. Perpendicular lines have gradients that, when multiplied together equal -1. (i.e. If m 1 is the gradient of a tangent, and m 2 is the gradient of a normal to that tangent, m 1 m 2 = 1) 2.3 Distance Between Two Points The distance between two points,p 1 : (x 1, y 1 ) and P 2 : (x 2, y 2 ), on a co-ordinate axis is calculated using the formula P 1 P 2 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 1

5 2.4 Distance Between a Point and a Line The shortest distance between a point P : (x 0, y 0 ) and a line L : ax+by +c = 0 will always be the perpendicular distance. The distance P N (where N is the point on L where the perpendicular line meets) can be calculated using the equation P N = ax 0 + by 0 + c a2 + b 2 3 Conic Sections The conic sections are a family of curves including parabolas, ellipses, circles and hyperbolas. 3.1 Expressing Curves in Cartesian and Parametric Form The Parabola The parabola with the y-axis as its line of symmetry has the Cartesian equation 4ay = x 2. This parabola can also be expressed by the parametric equations x = 2at, y = at 2. The parabola with the x-axis as its line of symmetry has the Cartesian equation y 2 = 4ax. This parabola can also be expressed by the parametric equations y = 2at, x = at The Ellipse The ellipse, which has centre (0, 0), has the Cartesian equation x2 a 2 + y2 = 1. This ellipse can also be expressed b2 by the parametric equations x = a cos θ, y = b sin θ The Circle The circle, which has centre (0, 0), has the Cartesian equation x 2 + y 2 = a 2. This circle can also be expressed by the parametric equations x = a cos θ, y = a sin θ The Hyperbola The hyperbola, which has centre (0, 0), has the Cartesian equation x2 a 2 y2 = 1. This hyperbola has asymptotes b2 at y = ± b x and can also be expressed by the parametric equations x = a sec θ, y = b tan θ. a 3.2 Quadratic Equations of the Conic Sections All conic sections can be written as an equation of the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. For example, a circle with centre (a, b), radius r has the Cartesian equation (x a) 2 + (y b) 2 = r 2, which can be multiplied out to give the equation x 2 2ax + y 2 2by + a 2 + b 2 r 2 = Finding Tangents and Normals to the Curves There are two ways that the tangent and normal to a curve can be found - by using the parametric equations of the curve, and by using y = mx + c. This will focus on the method using parametric equations Finding the Tangent and Normal to a Curve using Parametric Equations Bold type gives the general stages for finding the tangent and normal to any curve with known parametric equations. Italicised type shows an example of finding the tangent and normal to a parabola. 1. Ascertain the Cartesian and parametric equations of the curve. The Cartesian equation of the parabola is y 2 = 4ax and the parametric equations are y = 2at, x = at 2. 2

6 2. Use differentiation to find m - find dy using the chain rule. Firstly, differentiate y = 2at. y = 2at dy = 2a Now differentiate x = at 2. x = at 2 Now find dy. dy = 2at dy = dy dt dt = 2a 2at = 1 t Therefore m = 1 t 3. Use y y 1 = m(x x 1 ) to find the equation of the tangent. Substitute values for x, y and m into the equation y y 1 = m(x x 1 ) giving: y 2at = 1 t (x at2 ) ty 2at 2 = x at 2 ty = x + at 2, which is the equation of the tangent. 4. To find the equation of the normal, first the value of m must be determined. For two perpendicular lines (y = m 1 x + c 1 and y = m 2 x + c 2 ), m 1 m 2 = 1. Therefore m 2 = 1. Let m 1 the gradient of the tangent, 1 t = m 1. This means m 2 = 1 1 = t. t 5. Use y y 1 = m(x x 1 ) to find the equation of the normal. Substitute values for x, y and m into the equation y y 1 = m(x x 1 ) giving: y 2at = t(x at 2 ) y 2at = tx + at 3 y = tx + 2at + at 3, which is the equation of the normal. 3.4 Directrices, Foci and Eccentricity Conics can be described in terms of the locus of points whose distance to a fixed line D (the directrix) and the point F (Focus) are in constant ratio e (Eccentricity). N P(x, y) D F Figure 1: The relationship between the curve, the focus F and the directrix D 3

7 Consider P (x, y) such that P F P N = e, or P F 2 = e 2 P N 2. N is the point on the directrix to give the shortest distance PN. So we use P N 2 = the directrix, has equation ax + by + c = 0. (ax + by + c)2 a 2 + b 2, where D, Theorem: Given any D, F and e, the following statements hold for the locus of points P (x, y) such that P F = e P N. If e < 1, the conic is an ellipse. If e = 1, the conic is a parabola. If e > 1, the conic is a hyperbola. 3.5 Identifying a Conic Section by its Equation Remember all conic sections have an equation in the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, it is possible to identify which conic we have an equation for by calculating B 2 4AC. If B 2 4AC = 0, it is a parabola. If B 2 4AC < 0, it is an ellipse. If B 2 4AC > 0, it is a hyperbola. 4 Functions 4.1 Domains, Codomains and Ranges What can go into a function is the DOMAIN. What may possibly come out is the CODOMAIN. What actually comes out is the RANGE. The set D is the domain. The set R is the codomain/range. For a real function defined by a formula, the maximal domain is the largest subset of D. It is required that for every x, there is only ONE value of f(x). i.e. The function must be a ONE-TO-ONE function or a MANY-TO-ONE function. 4.2 Algebra with Functions It is possible to combine functions as if they were numbers, as long as they have a common domain. i.e. The following rules are all defined on D(f) D(g). (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) fg(x) = f(x)g(x) f f(x) (x) = g g(x) 4.3 Composite Functions These are usually known as function of a function. We write f(g(x)) = (f g)(x). The domain of f g is the set of x D(g) such that f(x) D(f). Composite functions are sometimes written as f 2 (x). 4

8 4.4 Odd and Even Functions If f(x) is an odd function, then f( x) = f(x). For example, f(x) = x 3 and f(x) = sin(x) are odd functions - when plotted on a graph, they are symmetric under rotation by π. If f(x) is an even function, then f( x) = f(x). For example, f(x) = x 2 and f(x) = cos(x) are even functions - when plotted on a graph, they are symmetric about the y-axis. Most functions are neither odd nor even, but any function f(x) can be decomposed into E(f(x)) + O(f(x)), where: E(f(x)) = f(x) + f( x) 2 and O(f(x)) = f(x) f( x)). 2 Example of Using Odd and Even Functions Suppose f(x) is an odd function and g(x) is an even function, then is (f g)(x) an even function, odd function or neither? Let f(x) = y, then f( x) = y. Let g(x) = z, then g( z) = z. Remember: (f g)(x) = f(g(x)) (f g)(x) = f(z) = y (f g)( x) = f(z) = y Therefore (f g)(x) is an EVEN function! 4.5 Periodic Functions A function f(x) is periodic if there is a T > 0 such that f(x + T ) = f(x) for all x. T is the period of f(x) if it is the smallest such T. Examples of periodic functions are sin(x), cos(x) and tan(x). 4.6 Monotonic Functions f(x) is monotonic increasing if f(x 1 ) f(x 2 ) for all x 1 < x 2. Examples of monotonic increasing functions are e x and x 3. f(x) is strictly monotonic increasing if f(x 1 ) < f(x 2 ) for all x 1 < x 2. f(x) is monotonic decreasing if f(x 1 ) f(x 2 ) for all x 1 > x 2. Examples of monotonic decreasing functions are e x and x 3 f(x) is strictly monotonic decreasing if f(x 1 ) > f(x 2 ) for all x 1 > x Trigonometric Functions f(x) (f(x)) 1 sin(x) cos(x) tan(x) csc(x) sec(x) cot(x) If a function can be written as y(x) = A sin(ωx + α) then y is a sinusoidal function of x with amplitude A, angular frequency ω and phase α. The period P can be calculated as P = 2π ω. 5

9 4.8 Hyperbolic Functions The Hyperbolic Sine The Hyperbolic Cosine sinh(x) = ex e x cosh(x) = ex + e x 2 2 Note that sinh(x) and cosh(x) are the odd and even parts of e x respectively. f(x) sinh(x) cosh(x) tanh(x) (f(x)) 1 csch(x) sech(x) coth(x) Osbourne s Rule All trigonometric identities become hyperbolic trigonometric identities, EXCEPT that the product of sines changes sign. For example: Circular Hyperbolic 4.9 Inverse Functions cos 2 (x) + sin 2 (x) = 1 cosh 2 (x) sinh 2 (x) = tan 2 (x) = sec 2 (x) 1 tanh 2 (x) = sech 2 (x) sin(2x) = 2 sin(x) cos(x) sinh(2x) = 2 sinh(x) cosh(x) Given that f(x) is a function with domain D and range R, the inverse function f 1 (x) would have domain R and range D - i.e. f 1 (x) has the domain of f(x) as the range and vice versa. The inverse function will only exist if f(x) is a ONE TO ONE function. Inverse functions have the following property. (f f 1 )(x) = (f 1 f)(x) = x Finding the Inverse Function There are two methods for finding the inverse of a function: the graphical method and the algebraic method. Graphical Method To find the inverse of a function graphically, reflect the function in the line y = x. Algebraic Method To find the inverse of a function using algebra, use the following steps. 1. If necessary, find a subset of the domain on which the map is ONE TO ONE. 2. Solve y = f(x) for x in terms of y. 3. Interchange x and y to obtain y = f 1 (x). When the function is not one to one on the domain, we need to restrict the function to a domain (i.e. sub-domain) in which it IS one to one. a Inverse Trigonometric Functions As only one to one functions have an inverse, trigonometric functions as they stand do not have inverses. Therefore we must restrict the domain of a trigonometric function in order to find its inverse. The following table gives the domains on which it is possible to compute inverse functions for the associated trigonometric function. 6

10 Function Domain Range cos(x) R [-1, 1] arccos(x) [-1, 1] [0, π] sin(x) R [-1, 1] arcsin(x) [-1, 1] [ π 2, π 2 ] tan(x) R R arctan(x) R ( π 2, π 2 ) cot(x) R R arccot(x) R (0, π) These standard domains can be used to find inverses for other domains Inverse Logarithmic Functions If y = a x, then we can say that x = log a y (the logarithm of y to the base a). If y = e x, then we can say that x = log e y = ln y (the natural logarithm of y). Note that a x = e x ln a and therefore log a y = ln y, which is the change of base rule. ln a 4.10 Piecewise Functions Piecewise functions are functions that have different formulae in different regions of the domain. 0 x 1 For example, f(x) = 1 x 2 1 < x < 1 is a piecewise function. x x 1 5 Sequences, Series and Limits 5.1 Sequences A sequence is simply a list of real numbers - a 1, a 2, a 3,... = {a n }. a n is a general term in the sequence - the n th term. a 1 is the first term in the sequence. An infinite sequence is one that continues indefinitely. A finite sequence is one with a finite number of terms. 5.2 Defining Sequences It is possible to specify a series in a number of different ways: 1. By specifying a general term: Usually known as the n th term - for example, a general term of a sequence may look like n 2n + 1. The general term should also specify the range of n. 2. By giving a recursion formula: For a sequence a 1, a 2, a 3,..., a recursion formula is one that requires the computation of all previous terms in order to find the value of a n. For example, a recursion formula may look like a n+1 = 5a n By specifying the sequence using other means: For example, the sequence {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5} are the digits of π. It is sometimes possible to express a recursion formula as a formula for a general term. For example: However this is NOT always possible. a n+1 = 5a n 2n + 1 with a 1 = 1 can be expressed as a n+1 = 2n ( 5) n (n!) (2n + 1)! 7

11 5.3 Convergence of Sequences We say that an infinite sequence, a n, converges to a limit L if for every ε (ε - epsilon, a very small amount) there is an N such that L ε < a n < L + ε for all n > N. In such a case, we write a n L as n or equivalently lim a n = L. n can say the sequence diverges or alternatively the limit does not exist. If there is no such L, then we Monotonic and Bounded Sequences We can say that a sequence (finite or infinite) is: Strict increasing if a n < a n+1 for all n Strict decreasing if a n > a n+1 for all n Increasing if a n a n+1 for all n Decreasing if a n a n+1 for all n A sequence is bounded above if there is a number M (known as an upper bound) such that a n M for all n N. Similarly, a sequence is bounded below if there is a number M (known as an lower bound) such that a n M for all n N. Some Important Remarks If a n is bounded, this does not imply that a n has a limit. If a n has a limit, this does imply that a n is bounded. If a n is unbounded, this implies that a n has NO limit. Theorem: If an infinite sequence is increasing and bounded above, then it must converge to a finite limit. Proof: There must be an infinite number of values of the sequence in a finite interval. being increasing, it cannot return to a value after passing it; hence it must have a limit. But owing to it The Sandwich Theorem Theorem: Suppose b n and c n converge to L as n and suppose that there is an N such that b n a n c n for all n N. Then a n L as n. The Sandwich theorem allows the computation of the limit of an expression by trapping the expression between two other expressions which have limits that are easier to compute Limit Laws for Sequences If lim n a n = A and lim n b n = B, with A, B R, then: lim n (a n ± b n ) = lim n a n ± lim n b n = A ± B lim n (a nb n ) = lim n a n a lim a n n n lim = n b n lim b = A n B n lim b n = AB n providing that B 0. If a n+1 = f(a n ) and a n L, then L = f(l) Infinite Limits If a sequence a n is such that a n > 0 for sufficiently large n and 1 a n 0 we say that a n. Similarly, if a n < 0 for sufficiently large n and 1 a n 0, we say that a n. In these cases, a n diverges to infinity. 8

12 5.3.5 Comparison Test for Sequences Suppose that a n and b n are sequences and there is an N such that a n b n for all n N. Then, if the limits exist, lim a n lim b n. Also, if a n, then b n. n n If a n does not converge, then it either diverges to ± or it oscillates (e.g. a n = ( 1) n oscillates) Standard Sequence Limits The following are standard sequence limits and may be quoted without proof: 1 1. If p > 0 then lim = 0 and lim n np n np =. 2. If r R and r < 1, then lim n rn = 0. If r R and r > 1, then If r R and r = 1, then lim n rn =. lim n rn = 1 (constant sequence). If r R and r = 1, then lim n rn diverges (oscillates). (ln n) a 3. If a R and b > 0, then lim n n b = 0. n a 4. If a R and p > 1 then, lim n p n = 0. c n 5. If c R, then lim n n! = 0. n q 6. If q R, then lim n n! = If a > 0, then lim a 1 n = 1 and moreover lim n 1 n = 1. n n 8. If x R, then lim n (1 + x n )n = e x. WARNING: Do not write lim n a n = b n as this does not make sense! 5.4 Series and The Sigma Notation For a general sequence a 1, a 2, a 3, a 4... a k 1, a k, a k+1..., the sum of the first n terms, a 1 + a 2 + a a n can be written That is to say, n a k. n a k = a 1 + a 2 + a a n. We let the index k in the term a k take, in turn, the values 1, 2, 3... n. These sums are called series. We can also sum infinite sequences, giving infinite series. a k = a 1 + a 2 + a An- other way of writing this would be a k = lim n n a k, where For example, n a k is called a partial sum. In order to use the Sigma notation, we must find a suitable expression to represent the general term a k. NOTE: Just as a limit may not exist, it is entirely possible that an infinite series may not be computable, in n particular if lim a k does not exist. In this case, the series is said to diverge - otherwise it converges. More n often than not, we can just work out whether a series converges or diverges, rather than actually computing the limit that the partial sums tend to. 9

13 5.5 Series and Partial Sums Consider the infinite series a k = a 1 + a 2 + a Obviously, it is not possible to calculate the sum of the entire series all at once. Therefore we can form a sequence of the sums of some terms where: S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3. S n = a 1 + a 2 + a a n So, S n is the sum of the first n terms of the infinite series - S n is called the nth partial sum of the series. An infinite series a k is convergent if the sequence of partial sums S 1, S 2, S 3, S 4... S k 1, S k, S k+1... in which S k = n a k is convergent. If not, the series is divergent. We define the sum of an infinite number of terms to be the limit of the sequence of partial sums as n. 5.6 Geometric Series Geometric series are of the form a + ar + ar ar n = ar n 1 where r can be positive or negative. In a geometric series, a is the first term, and r is the common ratio. For r 1 we can see whether a series converges using the formula S n = a(1 rn ). 1 r If r < 1, then r n 0 as n, leaving S n a 1 r If r > 1, then r n as n - the series diverges. - the series converges. If r = 1, then S n = a + a(1) + a(1) a(1) n 1 = na and lim n S n = ± - the series diverges. If r = 1, then the series diverges as the nth partial sum is either a or Arithmetic Series Arithmetic series are of the form a + (a + d) + (a + 2d) (a + (n 1)d) = a + (n 1)d where each term differs from the previous term by the same amount, d. In an arithmetic series, a is the first term, and d is the common difference Summing Arithmetic Series n n k = 1 n(n + 1) 2 a + (k 1)d = n (2a + (n 1)d) 2 For higher powers: n k 2 = 1 n(n + 1)(2n + 1) 6 n k 3 = 1 4 n2 (n + 1) Properties of Convergent Series n (a k + b k ) = n a k + n b k If both series converge (absolutely), 10

14 n (a k b k ) = n a k n BUT n c(a k ) = c n n a k (a k b k ) n n b k. a k b k 5.9 Standard Series Limits An infinite series a n converges if the sequence of partial sums S k = The following are standard series limits and may be quoted without proof: 1. If a n = 1 n p, then a n converges for p > 1 and diverges otherwise. 2. If a n = r n, then a n converges for r < 1 but diverges otherwise. We say that a n converges absolutely if a n converges, and if a n converges absolutely, then it converges. However, there are series that converge but do not converge absolutely Tests for Convergence and Divergence of a Series k a n has a limit as k. Often, we cannot find the limit of the nth partial sum as n. However we can determine whether the series will converge or diverge using one or some of the following tests Divergence Test If a n 0 as n, then a n diverges. NOTE: If a n 0 as n, then the sequence may converge or diverge Leibniz Theorem (The Alternating Series Test) This theorem states that ( 1) n a n will converge if the following three conditions are satisfied. 1. a n 0 for all n. 2. a n is a monotonically decreasing sequence - that is a n+1 a n for all n. 3. a n 0 as n Ratio Test Consider the limit L = lim a n+1 n a n. If L converges and L < 1, then a n converges. If L converges and L > 1, then a n diverges. If L does not converge, there is no conclusion. If L = 1, then there is no conclusion Comparison Test For sequences a n and b n : If the series b n diverges, and a n b n 0 for all n, then a n diverges. if the series b n converges, and b n a n 0 for all n, then a n converges. 11

15 Order In Which To Use Tests Unless it is obvious which test to use, apply the tests in the following order: 1. Divergence Test 2. Leibniz Theorem 3. Ratio Test 4. Comparison Test 5.11 Power Series A power series is a sum of terms which contains an unknown variable x raised to a power - a example of this is the binomial series expansion. A power series has the general form a 0 + a 1 x + a 2 x 2 + = a n x n. The convergence of a power series depends on the values of x chosen The Radius of Convergence For a power series, there exists a number R, known as the radius of convergence, such that: If x < R, the series is absolutely convergent. If x > R, the series is divergent. n=0 At x = R and x = R, the series may be convergent or divergent Calculating the Radius of Convergence For a power series a n x n, the radius of convergence can be calculated using the ratio test. This gives the n=0 result that R = lim a n n a n The General Power Series The general power series at x = x 0 is a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) 2 + = a n (x x 0 ) n. The same procedure for finding the radius of convergence applies - in this case: If x x 0 < R, the series is absolutely convergent. If x x 0 > R, the series is divergent. At x x 0 = R, the series may be convergent or divergent. 6 Limits and Continuity of Functions 6.1 Definition of a Limit For a function f(x), if the values of f(x) can be made as close as possible to L, by taking values of x sufficiently close to a (but not equal to a), then write lim n a f(x) = L. NOTE: It is not necessary for the function to be defined at a to have a limit there. 6.2 Left and Right Limits If lim f(x) = L converges, then it can be said that lim f(x) = lim f(x) = L. n a n a n a + Also, if lim f(x) lim f(x), then lim f(x) does not exist. n a n a + n a These rules can be used to determine whether a function is continuous at x = a, especially with piecewise functions. n=0 12

16 6.2.1 Limit Laws for Functions Let a R, lim n a f(x) = F and lim n a g(x) = G, then: lim n a f(x) ± g(x) = lim n a f(x) ± lim n a g(x) = F ± G lim n a f(x)g(x) = [ lim n a f(x)][ lim n a g(x)] = F G lim [ f(x) lim f(x) n a g(x) ] = n a lim g(x) = F G n a 7 Differentiation providing G Differentiation by First Principles f(x + h) f(x) If f(x) is defined in some interval containing x, then is the gradient of the line passing through h points (x, f(x)) and (x + h, f(x + h)). If we take the limit as h 0, then we obtain the differential of f(x) with respect to x at x = a, if the limit exists. Thus, df = lim h 0 f(x + h) f(x). h 7.2 Continuity and Differentiation If f(x) is differentiable at x = a, then f(x) is continuous at x = a. There are two main ways that a function can be continuous, yet not differen- The converse is NOT true. tiable at x = a. At corner points. At points of vertical tangency. At these points the gradient of the tangents have different limits from the left and right, and thus the limit defines that the derivative does not exist. 7.3 Rules of Differentiation Standard Differentiation Results For any a R: d xa = ax a 1 d ln(ax) = 1 x d eax = ae ax d sin ax = a cos ax d cos ax = a sin ax d tan ax = a sec2 ax d arcsin x a = 1 a2 x 2 d arccos x a = 1 a2 x 2 13

17 d arctan x a = a a 2 + x 2 d sinh ax = a cosh ax d cosh ax = a sinh ax d ax = ln a a x Other Rules for Differentiation For the two functions u(x) and v(x): Linearity Rule: Product Rule: Quotient Rule: If y = y(u) and u = u(x): Chain Rule: d(u + v) d(uv) = du + dv = v du + u dv d u du v v = u dv v 2 dy = dy du du 7.4 Implicit Differentiation There are two ways to define a function, explicitly and implicitly. Explicit definition of a function: These can be written as y = f(x). For example, y = 2x + 3 is an explicit equation. Implicit definition of a function: These can be written as f(x, y) = a where a is any number, possibly 0. For example y 2x = 3 is an implicit definition of y as a function of x. Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. Consider the implicit function f(x, y) = 0. When differentiating this function, let y be a function of x. Then differentiate with respect to x giving an equation in x, y, and dy dy. Finally rearrange to make the subject. 7.5 Computing Higher Derivatives This involves repeated differentiation of a function. d y = dy ( ) dy d d etc. ( d 2 y 2 = d2 y 2 ) = d3 y Parametric Differentiation We can differentiate x = g(t) to give dg dt Then dy df = dt = df dt dt dg. dg dt and y = f(t) to give df dt. Also, by chain rule, we can compute higher derivatives - d2 y 2 = d dt 14 ( ) df dt dt dg dt

18 7.7 Logarithmic Differentiation If y = a x, write this as ln y = x ln a and then differentiate this implicitly. 7.8 Differentiation of Inverse Functions If y = f(x), then let g = f 1. Then g(f(x)) = x, and so g (f(x))f (x) = 1. This implies that g (y)f (x) = 1. This gives the formula dy = ( ) 1 dy = 1. dy 7.9 Leibniz Rule for Repeated Differentiation of Products Suppose y(x) = f(x)g(x) (i.e. two functions multiplied together), then dn y n = n 8 Applications of Differentiation 8.1 Linearization r=0 ( n r ) f (r) g (n r). Differentiation can be used to find the best fit straight line to any curve. For example, if y = f(x), then the equation of a tangential line at x = a is given by y f(a) = f (a). x a Therefore the function L(x) = f (a)(x a) + f(a) is the linearization of f(x) at x = a. The linearization of any straight line is simply the straight line itself - this is the only function with this property. 8.2 Extrema of Functions The extrema of a function is the maxima (maximum points) and minima (minimum points) Extreme Value Theorem for Continuous Functions Theorem: If f(x) is continuous at every value in the interval [a, b], then f takes on both its maximum and minimum values in this interval. In other words, there are points x max [a, b] and x min [a, b] such that f(x min ) f(x) f(x max ) for all x [a, b]. If the interval is not closed, or is infinite, then this theorem does not hold Local Extrema Local extrema comprise of local maxima and local minima. A local maximum is a maximum point within a given interval - it need not be the global maximum. Similarly, a local minimum is a minimum point within a given interval Derivatives and Extrema Suppose a function f(x) is continuous and differentiable for x [a, b]. If there is a local extremum at a point c such that a < c < b, then f (c) = 0. If there is no such point, then the function is locally monotonic for x [a, b]. 8.3 Convexity and Concavity A function f is said to be concave if any chord joining two points on the curve lies above the curve. Similarly, if any chord joining two points lies below the curve, the function is said to be convex. If f (x) < 0 for x [a, b] then the function is concave for the interval [a, b]. If f (x) > 0 for x [a, b] then the function is convex for the interval [a, b]. 15

19 8.3.1 Critical Points The critical points of a function f(x) are defined to be the set of points where: f (x) = 0 f (x) is undefined. Note that not all critical points are extrema. Classification of Critical Points where f (x) = 0 If f (x) > 0 then x is a local minimum. If f (x) < 0 then x is a local maximum. If f (x) = 0 and f (x) = 0, then the critical point may be a local maximum, local minimum or neither. If f (x) = 0 and f (x) 0, then x is a point of inflexion, where the function changes from concave to convex, or vice versa. Note, in this case f (x) need not equal zero. 8.4 Limits using Calculus - L Hôpital s Rule It is possible to determine the limit of a ratio of two functions whose limits are both zero using calculus. If the two functions are differentiable, then we can apply L Hôpital s Rule, which states that f(x) lim x a g(x) = lim f (x) x a g (x). L Hôpital s Rule can be used successively to evaluate the limit of a ratio of two functions as it follows that f(x) lim x a g(x) = lim f (x) x a g (x) = lim f (x) x a g (x) =... etc. 8.5 Intermediate Value Theorem Theorem: A function that is continuous for x [a, b] will take on every value between f(a) and f(b) Finding the Root of an Equation The intermediate value theorem tells us that if a function is continuous for x [a, b], then every interval in which f(x) changes sign must contain a root of the equation f(x) = Rolle s Theorem Theorem: Suppose that f(x) is continuous for x [a, b] and is differentiable for x (a, b) and f(a) = f(b), then there is at least one value c [a, b] where f (c) = 0. If f(x) is constant (i.e. such as the line y = 4) then clearly f (c) = 0 for all a < c < b. If f(x) is not constant, then f(x) must attain its extrema at some a < c < b - this can be shown using the extreme value theorem. For this theorem to be applicable, it is crucial that the function is differentiable for the interval specified. 8.7 Mean Value Theorem Theorem: Suppose that f(x) is continuous for x [a, b] and is differentiable for x (a, b), then there is at f(b) f(a) least one value of c (a, b) such that = f (c). b a Corollaries of the Mean Value Theorem If f (x) = 0 for x (a, b) then f(x) is constant. The Constant Difference Theorem Theorem: If f (x) = g (x) at each x (a, b) then f(x) = g(x) + c. The constant difference theorem has a simple geometric interpretation. It tells us that if f and g have the same derivative for an interval then there is a constant c such that f(x) = g(x) + c for each x in the interval. That is, the graphs of f and g can be obtained from one another by a vertical translation of c. 16

20 8.8 Maclaurin Series If a function f(x) can be differentiated n times at x = 0, then the nth term of the polynomial for the Maclaurin Series expansion for f(x) at x = 0 is p n (x) = f(0) + f (0)x + f (0)x 2 + f (0)x f (n) (0)x n = n f (k) (0)x k. 2! 3! n! k! The Radius of Convergence The convergence set for a power series of x is called the interval of convergence. If the convergence set is: The single value x = 0, we say that the radius of convergence is 0. The interval (, ), we say that the radius of convergence is +. The interval ( R, R), we say that the radius of convergence is R. 8.9 Taylor Series Whilst the Maclaurin Series finds a series expansion for x = 0, the Taylor Series can be used to find a series expansion at x = a. If a function can be differentiated n times at x = a, then the then the nth term of the polynomial for the Taylor Series expansion for f(x) at x = a is p n (x) = f(a) + f (a)(x a) + f (a)(x a) 2 + f (a)(x a) f (n) (a)(x a) n = n f (k) (a)(x a) k. 2! 3! n! k=0 k! If the expansion has no index n, at which the polynomial is stopped, the result is the Taylor series expansion for f(x) - i.e. the series is infinite. k= The Radius of Convergence For any power series with terms in (x - a), exactly one of the following must be true. 1. The series converges only at x = a. In this case the radius of convergence is The series converges absolutely for all real values of x. In this case, the interval of convergence is (, ) and the radius of convergence is The series converges absolutely for all real values of x in some open interval (a R, a + R), and diverges for all values outside this interval. For values x = a R and x = a+r, the series may converge or diverge. In this case, the interval of convergence is (a R, a + R), and the radius of convergence is R The Remainder Theorem of a Taylor Polynomial We can find a measure of the accuracy of approximating a function f(x) to its nth Taylor polynomial p n (x). We use the idea of a remainder, giving f(x) = p n (x) + R n (x), where the value R n (x) is the error associated with the approximation. Finding a bound for R n (x) gives an indication of the accuracy of the approximation f(x) p n (x) Taylor s Theorem Theorem: If the function f is differentiable n + 1 times in an open interval I containing a, then for each x in I, there exists a number c between x and a such that f(x) = f(a) + f (a)(x a) + f (a)(x a) 2 + f (a)(x a) f (n) (a)(x a) n + R n (x) 2! 3! n! where R n (x) = f n+1 (c) (n + 1)! (x a)n+1. Taylor s theorem is a generalisation of the mean value theorem. If R n (x) 0 as x, for all x I, we say that the Taylor series generated by the function f at x = a converges to the original function f for the interval I, and therefore we can state that f(x) = f (k) (a)(x a) k - i.e. the remainder tends to zero so it k=0 k! need not be included. 17

21 8.9.4 The Remainder Estimation Theorem Theorem: If there is a positive constant M such that f (n+1) (t) M for all t between x and a inclusive, then x a n+1 the remainder term R n (x) in Taylor s Theorem satisfies the inequality R n (x) M. (n + 1)! If this condition holds for every n and the other conditions of Taylor s Theorem are satisfied by f, then the series converges to f(x). 9 Curve Sketching This is a list of criteria to take into consideration when sketching a curve. Points where y = 0 Points where x = 0 Local maxima and minima (turning points where dy = 0) If dy > 0, the function is increasing If dy < 0, the function is decreasing Points of inflexion, where d2 y 2 = 0 but d3 y 3 0 Odd/Even function Periodic function Limits as x ± Limits as y ± Asymptotes (are there any restrictions on values for x and y?) Symmetry These may be considered in any order. 18

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