Antiderivatives the F T C and differential equations Introduction Goals Learn how to find antiderivatives integrals using Mathematica how to solve differential equations and how to understand the fundamental theorem of calculus from a Mathematica perspective Examples Antiderivatives Mathematica is pretty good at finding antiderivatives Integrate x 3 x x4 4 We can verify the correctness of an antiderivative by taking its derivative remember that refers to the most recent Mathematica output i e here this is the result of the Integrate command Integrate x 3 x D x x4 4 x3 Mathematica says that 1 4 x4 is an antiderivative of x3 The derivative of 1 4 x4 is x3 So this is correct The same thing but in one step D Integrate x 3 x x x3 The Fundamental Theorem of Calculus The first part of the Fundamental Theorem states that for any continuous function f on a b the function F x a f t t is differentiable on a b and F x x d f t t dx a f x x In other words if we have a definite integral with variable upper limit then the rate of change with respect to the upper limit is precisely the value of the integrand evaluated at the upper limit 2 Antiderivatives the FTC and differential equations nb The second part of the Fundamental Theorem states that for every continuous function f on a b a f t t F b F a b where F is some antiderivative of f Thus this gives a way to evaluate definite integrals Note that we are using to define our function instead of This speeds up the performance of the animation below quite significantly We define an interval a b and a function f with which we want to illustrate the Fundamental Theorem Moreover for the animations below we also find the maximum and minimum values for f on a b a 1 b 3 3 x 1 2 max MaxValue f x a x b x min MinValue f x a x b x f x 1 3x Abs x Abs x 2 a f x x the left hand side of the Fundamental Theorem First we use Mathematica to directly compute the integral b lhs Integrate f x x a b 2 3 2 3 3 Next we compute the function F x a f t t We add the assumption x a to prevent Mathematica from going off and doing x too general calculations with complex functions intf x Integrate f t t a x Assumptions x a a f t t F b F a And now we check if b lhs intf b intf a True The following procedure produces an animation which illustrates the Fundamental Theorem x We will use the derivative F of the function F x a f t t which we called inft above By moving the slider you change the value of x The information displayed during the animation is the increment Dx we use when going from one frame to the next x the current area under the graph i e F x a f t t the previous area i e for x D x F x F x D x Dx the exact rate of change given by F x and also as guaranteed by the Fundamental Theorem i e f x the approximate rate of change of the area i e Antiderivatives the FTC and differential equations nb 3 dx 0 1 Manipulate Column Row increment dx Row previous area N intf x dx Row current area N intf x Row approx rate of change 1 dx N intf x N intf x dx Row F x N intf x Row f x N f x Plot f t t a x Filling Axis PlotRange a b min max x a dx b dx x increment 0 1 previous area 0 659639 current area 0 958725 approx rate of change 2 99086 F x 2 88666 f x 2 88666 Here s another animation tat is a little fancier It also allows you to change the increment Dx with a slider Note that the animation runs into problems i e produces out of range results when x D x becomes too small To prevent it from crashing we ve added the If statement at the beginning of the code Observe how the different rate of change values become closer as D x is chosen smaller You could make the same observation with the previous Manipulate if you manually change the value of the variable dx outside Manipulate 4 Antiderivatives the FTC and differential equations nb Manipulate If x a dx x a 1 001 dx Column Row previous area N intf x dx Row current area N intf x Row approx rate of change 1 dx N intf x N intf x dx Row F x N intf x Row f x N f x Show Plot f t t a x dx Filling Axis PlotRange a b min max Plot f t t x dx x Filling Axis FillingStyle Directive Opacity 0 5 Red PlotRange a b min max PlotStyle Red x 1 a b dx dx 0 1 b a 10 5 b a 2 x dx previous area 3 22466 current area 3 33333 approx rate of change 0 590499 F x 0 5 f x 0 5 Differential Equations A differential equation is an equation which involves an unknown function along with some of its derivatives The solution of a differential equation is a function A Simple Model for Free Fall Newton s second law states that force is equal to mass times acceleration where acceleration is the change in velocity i e F m dv dt Near the earth s surface the force due to gravity is the weight of an object i e F mg If we assume we are in a vacuum and no other forces are acting then balancing these two forces leads to the simple differential equation for the free fall velocity of a body m dv dt m g or dv dt g The solution found by straightforward integration of this model is v t dv dt dt g dt g t C Antiderivatives the FTC and differential equations nb 5 And if the velocity at t 1 is equal to g i e v 1 g then C 0 and v t g t In addition to using integration we can solve also much more complicated differential equations in Mathematica using the builtin DSolve command soln DSolve v t g v 1 g v t t v t g t We can also assign the value of g into the solution soln g 9 81 v t 9 81 t Without using DSolve we could integrate the equation dv dt dt g dt from above by hand i e soln Solve Integrate v t t Integrate g t v t soln g 9 81 v t g t v t 9 81 t
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