Definition of the Derivative Function

Definition of the Derivative Function: f +xh/  f +x/ f ' (x) = limh‘0 cccccccccccchcccccccccccccccc Definition of Derivative at a Point: f +ah/ f +a/ f ' (a) = limh‘0 cccccccccccccccc ccccccccccccc (note: the first definition results in a function, the second definition results in a number. Also h f +ah/ f +a/ note that the difference quotient, cccccccccccccccc ccccccccccccc , by itself, represents the average rate of change of f from x = a to x = a + h) h Interpretations of the Derivative: f ' (a) represents the instantaneous rate of change of f at x = a, the slope of the tangent line to the graph of f at x = a, and the slope of the curve at x = a. Derivative Formulas: (note:a and k are constants) d cccc cc +k/ dx 0 d cccc cc (k·f(x))= k·f ' (x) dx d cccc cc + f +x//n dx n+ f +x//n1  f ' +x/ d cccc cc [f(x) ± g(x)] = f ' (x) ± g ' (x) dx d cccc cc [f(x)·g(x)] = f(x)·g ' (x) + g(x) · f ' (x) dx d cccc cc , ccccf c+x/ ccccc 0 dx g+x/ g+x/ f ' +x/  f +x/g ' +x/ cccccccccccccccccccccccccccccccc ccccccccccccccccc +g+x//2 d cccc cc sin(f(x)) = cos (f(x)) ·f ' (x) dx d cccc cc cos(f(x)) = -sin(f(x))·f ' (x) dx d cccc cc tan(f(x)) = sec2 + f +x// º f ' +x/ dx d cccc cc ln(f(x)) = ccccf c1+x/ ccccc º f ' +x/ dx d cccc cc e f +x/ dx d cccc cc a f +x/ dx e f +x/ º f ' +x/ a f +x/ º ln a º f ' +x/ d cccc cc sin1  f +x/ dx f ' +x/ cccccccccccccccc cccccccc cccc r  2 d cccc cc cos1  f +x/ dx f ' +x/  cccccccccccccccc cccccccc cccc r  2 1+ f +x// 1+ f +x// d cccc cc tan1  f +x/ dx d cccc cc + f 1 +x// at x dx f ' +x/ cccccccccccccccc cccccc 1+ f +x//2 1 f +a/ equals cccccccc ccccc at x f '+x/ a L'Hopitals's Rule: f +x/ If limx‘a cccc cccccc g+x/ f +x/ limx‘a cccc cccccc g+x/ f ' +x/ ˆ cccc00 or cccc cc and if limx‘a cccccccc ccccc exists then ˆ g ' +x/ f ' +x/ limx‘a cccccccc ccccc g ' +x/ f +x/ ˆ The same rule applies if you get an indeterminate form ( cccc00 or cccc cc ) for limx‘ˆ cccc cccccc as well. ˆ g+x/ Slope; Critical Points: Any c in the domain of f such that either f ' (c) = 0 or f ' (c) is undefined is called a critical point or critical value of f. Tangents and Normals The equation of the tangent line to the curve y = f(x) at x = a is y - f(a) = f ' (a) (x - a) The tangent line to a graph can be used to approximate a function value at points very near the point of tangency. This is known as local linear approximations. Make sure you use ž instead of = when you approximate a function. The equation of the line normal(perpendicular) to the curve y = f(x) at x = a is 1 y - f(a) =  cccccccc ccccc +x  a/ f ' +a/ Increasing and Decreasing Functions A function y = f(x) is said to be increasing/decreasing on an interval if its derivative is positive/negative on the interval. Maximum, Minimum, and Inflection Points The curve y = f(x) has a local (relative) minimum at a point where x = c if the first derivative changes signs from negative to positive at c. The curve y = f(x) has a local maximum at a point where x = c if the first deivative changes signs from positive to negative. The curve y = f(x) is said to be concave upward on an interval if the second derivative is positive on that interval. Note that this would mean that the first derivative is increasing on that interval. The curve y = f(x) is siad to be concave downward on an interval if the second derivative is negative on that interval. Note that this would mean that the first derivative is decreasing on that interval. The point where the concavity of y = f(x) changes is called a point of inflection. The curve y = f(x) has a global (absolute) minimum value at x = c on [a, b] if f(c) is less than all y values on the interval. Similarly, y = f(x) has a global maximum value at x = c on [a, b] if f (c) is greater than all y values on the interval. The global maximum or minimum value will occur at a critical point or one of the endpoints. Related Rates: If several variables that are functions of time t are related by an equation (such as the Pythagorean Theorem or other formula), we can obtain a relation involving their (time)rates of change by differentiating with respect to t. Approximating Areas: It is always possible to approximate the value of a definite integral, even when an integrand cannot b be expressed in terms of elementary functions. If f is nonnegative on [a, b], we interpret ¼a f +x/Å x as the area bounded above by y = f(x), below by the x-axis, and vertically by the lines x = a and x = b. The value of the definite integral is then approximated by dividing the area into n strips, approximating the area of each strip by a rectangle or other geometric figure, then summing these approximations. For our discussion we will divide the interval from a to b into n strips of equal width, 'x. The four methods we learned this year are listed below. Left sum: ½n1 i 0 f +ti /'t = f +t0 /'t+ f +ti /'t+ f +t2 /'t+ · · ·  f +tn1 /'t, using the value of f at the left endpoint of each subinterval. Right sum: ½ni 1 f +ti /'t = f +t1 /'t+ f +t2 /'t+ f +t3 /'t+ · · ·  f +tn /'tusing the value of f at the right endpoint of each subinterval. ti ti1 Midpoint sum: ½ni 0 f + cccccccc cccccc /'t 2 each subinterval. t0 t1 t1 t2 tn1 tn f + cccccccc cccc /'t  f + cccccccc cccc /'t  º º º  f + cccccccc cccccccc /'tusing the value of f at the midpoint of 2 2 2 Trapezoidal Rule: cccc12 + f +t0 /  f +t1 //'t  cccc12 + f +t1 /  f +t2 //'t  º º º  cccc12 + f +tn1 /  f +tn //'t. Note that the trapezoidal approximation is the average of the left and right sum approximations. Antiderivatives: The antiderivative or indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). Since the derivative of a constant is zero, the antiderivative of f(x) is not unique; that is, if F(x) is an integral of f(x), then so is F(x) + C, where C is any constant. Remember when you are integrating a function, f(x), you are finding a family of functions F(x) +C whose derivatives are f(x). Integration Formulas: ¼ k f +x/ Å x k¼ f +x/Å x ¼ # f +x/ ” g+x/'Å x ¼ un Å u 1 ¼ ccccu Å u n1 u cccc cccccc  C n1 ln « u « C ¼ cos uÅ u ¼ sin uÅ u ¼ f +x/Å x ” ¼ g+x/Å x sin u  C cos u  C ln « sec u « C ¼ tan uÅ u ¼ sec2 uÅ u ¼ eu Å u tan u  C eu  C ¼ au Å u u a cccc ccccc  C ln a c1ccccccc r cccc c Å u ¼ cccccccc a2 u2 1 cccccc Å u ¼ cccccccc a2 u2 sin1  ccccua  C cccc1a tan1  ccccua  C The Fundamental Theorems The First Fundamental Theroem of Calculus states If f is continuous on the closed interval [a, b] and F ' = f, then, ¼a f +x/Å x b F+b/  F+a/ The Second Fundamental Theorem of Calculus States If f is continuous on [a, b], then the function F(x) = ¼a f +t/Å t x has a derivative at every point in [a, b]. and d F ' (x) = cccc cc  f +t/Å t dx ¼a f +x/ x Definite Integral Properties (in addition to the indefinite integral properties) 1. ¼a f +x/Å x a 2. ¼a f +x/Å x b 3. ¼a f +x/Å x b 0  ¼b f +x/Å x a ¼a f +x/Å x  ¼c f +x/Å x c b Areas: If f(x) is positive for some values of x on [a, b] and negative for others, then b ¼a f +x/Å x represents the cumulative sum of the signed areas between the graph of y = f(x) and the x-axis (where the areas above the xaxis are counted positively and the areas below the x-axis are counted negatively) Thus, ¼a « f +x/ « Å x b represents the actual area between the curve and the x-axis. The area between the graphs of f(x) and g(x) where f(x) – g(x) on [a, b] is given by ¼a # f +x/  g+x/'Å x b Volumes: The volume of a solid of revolution (consisting of disks) is given by Volume = S¼left endpoint +radius/2 Å x+or dy/ right endpoint The volume of a solid of revolution (consisting of washers) is given by Volume = S¼left endpoint #+outside radius/2  +inside radius /2 'Å x+or dy/ right endpoint The volume of a solid of known cross-sectional areas is given by Volume = ¼left endpoint +cross sectional area/Å x+or dy/ right endpoint Arc Length: If the derivative of a function y = f(x) is continuous on the interval [a, b], then the length s of the arc of the curve of y = f(x) from the point where x = a to the point where x = b is given by b s=Ä a  dy 2  1  , cccc cc 0 Å x or à 1  + f '+x//2 Å x dx b a