Math Calculus Review Part 3

Cotangent Using the quotient rule (page 18) and the Pythagorean identity cos2 (x) + sin2 (x) = 1, the derivative of cotangent can be derived. cot(x) d [cot(x)] dx d [cot(x)] dx d [cot(x)] dx = = = = cos(x) sin(x) − sin2 (x) − cos2 (x) sin2 (x) −1 sin2 (x) − csc2 (x) Therefore, the derivative of cotangent is the negative of the square of cosecant. d [cot(x)] = − csc2 (x) dx Secant Using the quotient rule (page 18), the derivative of secant can be derived. sec(x) d [sec(x)] dx d [sec(x)] dx d [sec(x)] dx = = = = 1 cos(x) sin(x) cos2 (x) 1 sin(x) × cos(x) cos(x) sec(x) × tan(x) Therefore, the derivative of secant is secant multiplied by tangent. d [sec(x)] = sec(x) × tan(x) dx Cosecant Using the quotient rule (page 18), the derivative of cosecant can be derived. csc(x) d [csc(x)] dx d [csc(x)] dx d [csc(x)] dx = = = = 1 − sin(x) cos(x) − sin2 (x) 1 cos(x) − × sin(x) sin(x) − csc(x) × cot(x) Therefore, the derivative of cosecant is the negative of cosecant multiplied by cotangent. d [csc(x)] = − csc(x) × cot(x) dx Combining with Derivative Rules In most cases, one must determine the derivative of an an example that requires the use of derivative rules in addition to the knowledge of the derivatives of trigonometric function. One may apply the form trig (a) to many examples, where trig is the trigonometric function and a is the angle. d [trig])(a) × d [a]. Based on the chain rule (page 17), the derivative of trig (a) would be ( dx dx   d d d [trig (a)] = [trig](a) × [a], dx dx dx d [trig] is the derivative of the trigonometric function, and d [a] is the derivative of the angle. where dx dx Example: sin(2x) d [sin(2x)] dx d [sin(2x)] dx d [sin(2x)] dx  = =  d d [sin] (2x) × [2x] dx dx (cos(2x)) × 2 = 2 cos(2x) Asymptotes A linear asymptote is a straight line that a graph approaches, but does not become identical to. Asymptotes are formally defined using limits. See the the asymptotes section of the limits chapter on page 7 for more information. Figure 0.1: Graph demonstrating extrema on a curve Stationary Points Extrema Maxima and minima are points where a function reaches a highest or lowest value, respectively. A maximum occurs when positive slope changes to negative slope and a minimum occurs when negative slope changes to a positive slope. There are two kinds of extrema (a word meaning maximum or minimum): global and local, sometimes referred to as “absolute” and “relative,” respectively. A global maximum is a point that takes the largest value on the entire range of the function, while a global minimum is the point that takes the smallest value on the range of the function. Local extrema are the largest or smallest values of the function in the immediate vicinity. See Figure 0.1. All extrema look like the crest of a hill or the bottom of a bowl on a graph of the function. A global extremum is always a local extremum too, because it is the largest or smallest value on the entire range of the function, and therefore also in its vicinity. It is also possible to have a function with no extrema, global or local (e.g. y = x). At an extremum, the y-value is the value of the extremum and the x-value is where the extremum occurs. “Flatpoints” It is important to note that not all cases in which the first derivative of a function is equal to zero are turning points or extrema, though the first derivative of a function is equal to zero or does not exist at all turning points and extrema. “Flatpoints” (e.g. triple roots) may also occur when the first derivative of a function is equal to zero, but they are not turning points nor extrema because no slope change occurs. Classification At any extremum, the slope of the graph is zero or undefined, as the graph must stop rising or falling at an extremum, and begin to fall or rise. Because of this, extrema are also commonly called stationary points or turning points. If the graph has one or more of these stationary points, these may be found by setting the first derivative equal to zero and finding the roots of the resulting equation as well as values where the function is undefined. These values are referred to as critical points. Note that if the domain is restricted, the endpoints of the domain must also be checked to see if they are global extrema. critical point point in domain of f where f0 = 0 or f0 does not exist Extrema can only occur at critical points and endpoints. True extrema require a sign change in the first derivative. This makes sense — the graph must rise (positive first derivative) and fall (negative first derivative) to form a maximum. In between rising and falling, on a smooth curve, there will ideally be a point of zero slope — the maximum. A minimum would exhibit similar properties, but in reverse. First Derivative Test This leads to a simple method to classify a stationary point — plug x values (test points) slightly left and right into the derivative of the function. If the results have opposite signs then it is a true extremum. To calculate the coordinates of the minimum or maximum point, one would plug the determined x value into the original function to find its y value. • If f0 (x) < 0 for x < c and f0 (x) > 0 for x > c, then f(c) is a local minimum. • If f0 (x) > 0 for x < c and f0 (x) < 0 for x > c, then f(c) is a local maximum. Caution must be exercised with this method, as, if a point too far from the extremum is picked, one could take it on the far side of another extremum and incorrectly classify the point. A more rigorous method to classify a stationary point is called the extremum test that uses the second derivative, but this simple method is acceptable. Second Derivative Test • If f0 (c) = 0 and f00 (c) > 0, then c is a local minimum. • If f0 (c) = 0 and f00 (c) < 0, then c is a local maximum. Note that the second derivative test cannot be used to verify an extrema if the first or second derivative does not exist. Information Stationary Point First Derivative Second Derivative Minimum Point zero or undefined positive or undefined Maximum Point zero or undefined negative or undefined “Flatpoint” zero zero Figure 0.2: Graph containing an inflection point Stationary Point First Derivative Sign Before First Derivative Sign After Minimum Point negative positive Maximum Point positive negative “Flatpoint” same sign same sign Inflection Points Inflection points occur when the second derivative of a function is equal to zero. The curve changes from being concave up (positive second derivative) to concave down (negative second derivative), or vice versa. See Figure 0.2. “Flatpoints” are a specific type of inflection point where the graph flattens out (first derivative is zero), but the sign of the slope does not change. These points are called stationary points of inflection. Other inflection points are not “flatpoints,” and there is no flattening out (i.e. sine curve); these points are known as non-stationary points of inflection. Curvature Second Derivative First Derivative Graph Concave Up (“smile”) positive increasing Concave Down (“frown”) negative decreasing Inflection Point zero or undefined extrema Optimization Optimization is the use of Calculus in the real world. Calculus is a useful tool for maximizing or minimizing (also known as “optimizing”) a situation. Formulas Volume cube A = a3 , where a is the length of the side of each edge of the cube rectangular prism V = abc, where a, b, and c are the lengths of the 3 sides of the prism cylinder V = πr2 h, where r is the radius and h is the height of the cylinder 4 sphere V = πr3 , where r represents the radius of the sphere 3 Surface Area cube A = 6a2 , where a is the length of the side of each edge of the cube rectangular prism A = 2ab + 2bc + 2ac, where a, b, and c are the lengths of the 3 sides of the prism sphere A = 4πr2 , where r is radius of the sphere cylinder A = 2πr2 + 2πrh, where r is the radius and h is the height of the cylinder Addendum This section was designed for a test on derivatives administered by Jonathan Chernick to his AP4 Calculus BC class on October 14, 2008. It is not covered in Math 12H/4H. Alternative Definition of Derivative f0 (x) = lim x→a f(x) − f(a) x−a Parametric Equations Parametric equations are typically defined by two equations that specify both the x and y coordinates of a graph using a parameter. They are graphed using the parameter (usually t) to figure out both the x and y coordinates. The derivative of the parametrized curve x(t), y(t) is: dy dy dt , dx 6= 0 = dx dx dt dt Example: x = t, y = t2 dy = dx dy dt dx dt = 2t = 2t 1 Implicit Differentiation explicit relationship function in which f(x) is given in terms of x and constants; for every x-value there is one y-value implicit relationship relationship between two or more variables; two or more functions put together Ordinary differentiation is explicit differentiation. Implicit differentiation is useful when differentiating an equation that cannot be explicitly differentiated because it is impossible or hard to isolate variables (e.g. x2 + xy + y2 = 16). In many difficult problems involving implicit differentiation (e.g. multiple choice), it is necessary to substitute the dependent variable (e.g. y) and its derivatives (e.g. equation or previous determined derivative expressions. dy d2 y dx , dx2 ) based on the original Example: x2 + y2 = 1 Explicit Differentiation x2 + y2 y2 = 1 = 1 − x2 p y = ± 1 − x2 1 y = ±(1 − x2 ) 2 x dy = − dx y Implicit Differentiation x2 + y2 dy 2x + 2y dx dy 2y dx dy dx dy dx = 1 = 0 = = = −2x −2x 2y −x y Inverse Functions inverse function “opposite” of a function; if f(x) = a, f−1 (a) = f(x); reflected over line y = x The composition of a function and its inverse is x because the two functions “undo” each other.   f f−1 (x) = x With use of the chain rule (page 17), the relationship between the derivative of a function and the derivative of its inverse can be determined.   f f−1 (x) h i h i0 f0 f−1 (a) × f−1 (a) h i0 f−1 (a) = x = 1 1 = f0  f−1 (a)  A function and its inverse have reciprocal slopes with reversed (x, y) values. h f−1 i0 (a) = 1   f0 f−1 (a) h i0 Example: f(x) = x3 + x − 2, find f−1 (0) 0 = x3 + x − 2 x = 1 f0 (x) f0 (1) h f−1 h h h i0 f−1 i0 f−1 i0 f−1 i0 = 3x2 + 1 = 4 (a) = (0) = (0) = (0) = 1 f0  f−1 (a) f0  f−1 (0)  1  1 f0 (1) 1 4 Inverse Trigonometric Functions The inverse trigonometric functions are the inverse functions of the trigonometric functions. The inverse of the trigonometric functions sin, cos, tan, cot, sec, and csc is arcsin, arccos, arctan, arccot, arcsec, and arccsc, respectively. The notations sin−1 , cos−1 , etc. are often used for arcsin, arccos, etc., respectively, but this convention may result in confusion between multiplicative inverse and compositional inverse since this logically conflicts with the structure of expressions like sin2 x, which do not refer to function composition but rather multiplication. Each inverse trigonometric function has a derivative. Trigonometric Function Inverse (arc notation) Inverse (−1 notation) sin arcsin sin−1 cos arccos cos−1 tan arctan tan−1 cot arccot cot−1 sec arcsec sec−1 csc arccsc csc−1 In the table below, u can represent any differentiable expression, using the chain rule (page 17). Inverse Trigonometric Function arcsin u arccos u arctan u arccot u arcsec u arccsc u Derivative du , |u| < 1 dx du −1 √ × , |u| < 1 2 dx 1−u 1 du × 2 dx 1+u du −1 × dx 1 + u2 1 du √ × , |u| > 1 2 dx |u| u − 1 √ 1 1 − u2 × −1 du √ × , |u| > 1 2 dx |u| u − 1 Strategies for Simplifying In many difficult problems (e.g. multiple choice) where simplifying is necessary, there are some strategies for doing so. If simplifying is not required, these strategies are not necessary. • If an expression under an absolute value is always positive, the absolute value symbols can be removed. • Combine terms into terms with a common denominator. • Factor out variables from square roots. More Rules If the original expression is a constant raised to a variable power, use the cx rule (). If the original expression contains a variable in the base and exponent, logarithmic differentiation (page 31) must be used. ex The derivative of ex is itself. Based on the chain rule (page 17), where u is any differentiable expression, du d u [e ] = eu × dx dx cx c represents a constant. The derivative of cx is cx × ln c, c > 0 and c 6= 1. Based on the chain rule (page 17), where c is a constant, d u du [c ] = ln c × cu × , c > 0 and c 6= 1 dx dx ln x The derivative of ln x is x1 , x > 0. Based on the chain rule (page 17), where u is any differentiable expression, d 1 du [ln u] = × ,u>0 dx u dx Logarithms Properties These properties hold true for both log and ln. • log(xy) = log x + log y • log(x/y) = log x − log y • log xa = a ln x loga x = log x ln x = log a ln a Change of Base logb x The derivative of logb x is 1 . x ln(b) Based on the chain rule (page 17), where u is any differentiable expression, d 1 du [logb u] = × ; b > 0, b 6= 1, and u > 0 dx u ln(b) dx Logarithmic Differentiation Logarithmic differentiation is a differentiation process used to take the derivative of a variable raised to a variable or other complex situations. The natural log (ln) of both sides of an equation are taken, and the result is implicitly differentiated.