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AP CALCULUS NOTES SECTION 6.2 THE INDEFINITE INTEGRAL; INTEGRAL CURVES AND SLOPE FIELDS Some situations arise where you are asked to find a function F whose derivative is a known function f. For example, an engineer who can measure the variable rate at which water is leaking from a tank might want to know the total amount leaked over a certain period of time. Also, a biologist who knows the rate at which a bacteria population is increasing might want to know the size of the population in the near future. If such a function F exists, 1.) A function F is called an antiderivative of f on an interval I if F x f x for all x in I. Ex.1.) f x x4 Possible Antiderivatives: 1 F x x5 3 5 1 F x x5 1 5 1 F x x5 5 1 5 F x x 2 5 Ex.2.) f x cos x Possible Antiderivatives: F x sin x 4 F x sin x 2 F x sin x F x sin x 3 THEOREM: If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F x C where C is an arbitrary constant. Thus, the most general antiderivative of f on I is a family of functions F x C whose graphs are vertical translations of one another. A.) The Indefinite Integral: used to denote the collection of all antiderivatives of a function f. Indefinite Integral of f with respect to x: f x dx Integration – process of finding antiderivatives. d F x f x f x dx F x C dx Find the general antiderivative of each of the following functions: (Table on pg. 380) Ex.3.) f x 3x 2 Ex.4.) f x 3 x cos x 2 Ex.5.) f x 1 2x e x Ex.6.) f x 4sin x 2 x5 x x Antiderivative Linearity Rules: 1.) Constant Multiple Rule: Function kf x General Antiderivative kF x C, k is a constant 2.) Negative Rule: f x F x C 3.) Sum or Difference Rule: f x g x F x G x C In calculus, oftentimes you have to find a function given knowledge about its derivative B.) Differential Equations – an equation that involves a derivative of an unknown function. dy f x is called a differential equation dx The combination of a differential equation and an initial condition is called an initial value problem. By solving an initial value problem, we find the particular solution for y as opposed to the general solution y F x C . Ex.7.) Find f if f x 12x2 6x 4; f 1 1 . Ex.8.) Solve the initial value problem given dy 20 ex and y 0 2 . dx 1 x2 Ex.9.) A balloon ascending at a rate of 12 ft/s is at a height 80 ft above the ground when a package is dropped from the balloon. Given the acceleration of gravity is 32 ft/s 2 , how long does it take the package to reach the ground? C.) Slope Fields (The Geometry of Antiderivatives) – provide a visual of a family of antiderivatives (solutions of differential equations) Graphs of antiderivatives of a function f are called integral curves. All integral curves of a function f create a slope field. dy 2x . Ex.10.) Draw a slope field for the differential equation dx dy dy dy x, y x, y x, y dx dx dx 3, 2 1, 0 1, 2 3, 1 3, 0 3,1 3, 2 2, 2 2, 1 2,0 2,1 2, 2 1, 2 1, 1 1,1 1, 2 0, 2 0, 1 0, 0 0,1 0, 2 1, 2 1, 1 1,0 1,1 2, 2 2, 1 2, 0 2,1 2, 2 3, 2 3, 1 3,0 3,1 3, 2 Find the particular solution y f x to the given differential equation with the initial condition f 0 1 . Graph this particular integral curve on the slope field above. Note: all of the points that have the same x-coordinate will have the same slope since our differential equation has an x-term but no y-term. Ex.11.) Match each slope field with the differential equation each slope field could represent. A.) B.) C.) D.) dy x2 dx dy 2 2.) dx x dy cos x 3.) dx 1.) dy sin x dx dy ex 5.) dx dy 1 3 x 6.) dx 4 4.)