ࡱ> q`  bjbjqPqP .::''%vvvvvvvRQRQRQ8QtQh2RR" S S S S S Sggggggg$ih6lgv S S S S Sgvv S SVhpUpUpU Sdv Sv SgpU SgpUpU_chvvd SR eN^RQScKe4lh0hcmTm dmvd S SpU S S S S SggUX S S Sh S S S S$<@@vvvvvv FUNCTIONS INCLUDING EXPONENTIAL AND LOGARITHMIC FUNCTIONS The set of value on which the function acts is called the domain and the corresponding set of image values is called the range. f(x) = x + 2354657DomainRange The domain of a function may be an infinite set such as R. If for each element of y in the range, there is a unique value of x such that f(x) = y then f is a one-one function. If any element y in the range, there is more than one value satisfying f(x) = y then f is many-one. f(x) = 2x with domain ROne-onef(x) = x2 with domain RMany-onef(x) = x2 with domain x > 0One-onef(x) = sinx with domain 0 < x < 360Many-onef(x) = sinx with domain -90 < x < 90One-one COMPOSITION OF FUNCTIONS fgx!f(x)!gf(x) fg(x) and gf(x) are rarely the same. THE INVERSE OF A FUNCTION Usually written as f-1, that undoes the effect of f. f(x) = x + 2f-1(x) = x - 2 f-1f(x) = x. Many-one functions do not have an inverse as it is impossible to reverse them, unless you restrict its domain. For instance if f(x) = x2 had the domain of x > 0. THE MODULUS FUNCTION |x| = x when x > 0 or x when x < 0. The expression |x-a| can be interpreted as the distance between the numbers x and a on the number line. In this way the statement |x-a| < b means that the distance between x and a is less than b. It follows that a b < x < a + b. TRANSFORMING GRAPHS. Known FunctionNew FunctionTransformation y = f(x)y = f(x) + aTranslation through a units parallel to y-axis.y = f(x-1)Translation through a units parallel to x-axis.y = af(x)One-way stretch with scale factor a parallel to the y-axis.y = f(ax)One-way stretch with scale factor 1/a parallel to the x-axis. CO-ORDINATE GEOMETRY CARTESIAN AND PARAMETRIC FORM Cartesian Form A curve in Cartesian form is defined in terms of x and y only, for example y = x2 + 3x in which y is given explicitly in terms of x. x2 + 2xy + y2 = 9 in which the equation is given implicitly. Parametric Form A curve is defined in parametric form by expressing x and y in terms of a third variable, for example x = f(t), y = g(t). In this case, t is the parameter. y = 4t ! t = y/4 x = 2t2 ! x = 2(y/4)2 = y2/8 ! y2 = 8x When the parameter is , it may be necessary to use a trigonometric identity to find the Cartesian equation. For example, when x = acos  and y = asin , use cos2  + sin2  = 1. This gives: x2 + y2 = a2. This is a circle, centre (0,0), radius a. Also, when x = acos, y = bsin, x2/a2 + y2/b2 = 1. This curve is an eclipse. TRIGONOMETRY SECANT, COSECANT AND CONTANGENT secx = 1 / cosx cosecx =1 / sinx cotx = 1 / tanx Each function is periodic and has either line or rotational symmetry. The domain of each one muct be restricted to avoid division by zero. y =sec x. The period of sec is 360 to match the period of cos. Notice that secx is undefined whenever cosx = 0. The graph is symmetrical about every vertical line passing through a vertex. It has rotational symmetry of order 2 about the points in the x-axis corresponding to 90 180n. y = cosec x The period of cosec is 360 to match the period of sin. Notice that cosecx is undefined whenever sinx = 0. The graph is symmetrical about every vertical line passing through a vertex. It has rotation symmetry of order 2 about the points on the x-axis corresponding to 0, 180, 360, etc. y = cot x The period of cot is 180 to match the period of tan. Notice that cotx is undefined whenever sinx = 0. The graph has rotationalorder 2 about the points on the x-axis corresponding to 0, 90, 180, etc. INVERSE TRIGONOMETRIC FUNCTIONS The sine, cosine and tangent functions are all many-one and so do not have inverses on their full domains. However, it is possible to restrict their domains so that each one has an inverse. When you use the functions sin-1, cos-1, tan-1 on your calculator, the value given is called the principal value (PV). TRIGONOMETRIC IDENTITIES Pythagorean identities: cos2x +sin2x = 1 1 + tan2 x = sec2x cot2 x +1 = cosec2 x Compound angle identities: sin(A+B) = sinA cosB + cosA sinB cos(A+B) = cosA cosB sinA sinB tan(A+B) = tanA + tanB 1 tanA tanB Simply reverse the signs for (A-B). Double angle identities: sin 2A = 2sinA cosA cos 2A = cos2A sin2A = 2cos2A 1 = 1 2sin2A tan 2A = 2tanA 1- tan2A Half angle identities: sin2 A = (1 cosA) cos2 A = (1 + cosA) PROVING IDENTITIES The basic technique for proving a given identity is: Start with the expression on one side of the identity. Use known identities to replace some part of it with an equivalent form. Simplify the result and compare with the expression on the other side. DIFFERENTIATION THE CHAIN RULE dy = dy du Where u is a function of x. dx du dx The chain rule can also be used to establish results for connected rates of change. For example, the rate of change of the volume of a sphere can be written as dv/dt. The corresponding rate of change of the radius of the sphere can be written as dr/dt. Using the chain rule, the connection between these rates of chang is given by: dv = dv dr dt dr dt Also since v =4/3 r3, it follows that dv/dr = 4r2, so, the connection can now be written as dv/dt = 4r2dr/dt. EXPONENTIAL FUNCTION ex If y = ef(x) then dy/dx = f (x)ef(x). LOGARITHMIC FUNCTIONS ln x AND ln f(x) If y = ln x then dy/dx = 1 / x. If y = ln f(x) then dy/dx = f(x)/f(x) TRIGONOMETRIC FUNCTIONS d sin(ax+b) =a cos (ax +b) dy d cos(ax+b) =-a sin (ax +b) dy d tan(ax+b) =a sec2 (ax +b) dy THE PRODUCT RULE If y = uv, where u = f(x) and v = g(x), then dy/dx = u dv/dx + v du/dx. THE QUOTIENT RULE If y = u / v where u =f(x) and v = g(x), then dy/dx = v du/dx u dv/dx v2 USING THE RESULT dy/dx = 1 / dx/dy dy/dx = 1 / dx/dy = 1/dx/dy. If y = ax, then dy/dx = ax ln a. PARAMETRIC FUNCTIONS If x and y are each expressed in terms of a parameter t, then dy/dx = dy/dt dt/dx. Remember that dt/dx = 1 / (dx/dt). IMPLICIT FUNCTIONS To find dy/dx when an equation in x and y is given implicitly, differentiate each term with respect to x, remember that d f(y) /dx = df(y) /dy dy/dx. 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