THE COMPOSITE FUNCTION If, (a composite function) then In Words: Sub u in for t and multiply by u’ĭemonstration: Find: Verify In Words: Sub in for t and multiply by the derivative ofĮxample : Find without Integrating: If, solve forĮxample: Rewriting the Integral Find without integrating: Show middle stepĮxample: Rewriting the Integral - Two variable limits: Find withoutIntegrating: break into two parts. Explain the relationship between differentiation and integration. THE COMPOSITE FUNCTION If g(x) is given instead of x: In words: Substitute in g(x) for t and then multiply by the derivative of g(x)…exactly the chain rule (derivative of the outside * derivative of the inside) Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals. Since A(x) is a function, what then is the rate of change of that function? Take derivative In words, integration and differentiation are inverse operationsĢnd Fundamental Theorem of Calculus Given:, we want to find Note: a is a constant, u is a function of x and the order matters! “a” is a constant 2nd Fundamental Theorem of Calculus: If fis continuous on an open interval, I, containing a point, a, then for every x in I :ĭemonstration: find In Words: Sub in the function u and multiply by derivative of uĮxample: Find and verify: this Not this = The Definite Integral is actually finding points on the Accumulation graph. Evaluate at 1 Evaluate the Definite Integral for each of these points. The Definite Integral as a Particular Function: Evaluate the definite integral. Given an Initial Condition we find the Particular Function The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given. Ü Greeks created spectacular concepts with geometry, but not arithmetic or algebra very well.6034 Fundamental Theorem of Calculus (Part 2) AB CALCULUS
Ü And if you think Greeks invented calculus? No, they did not. It traveled as high up to its peak and is falling down, still the difference between its height at t=0 and t=1 is 4ft. Bear in mind that the ball went much farther. The height of the ball, 1 second later, will be 4 feet high above the original height. Therefore, if a ball is thrown upright into the air with velocity identify, and interpret, ∫10v(t)dt.Įxecuting the Second Fundamental Theorem of Calculus, we see Anie wins the race, but narrowly.Ī ball is thrown straight up from the 5 th floor of the building with a velocity v(t)=−32t+20ft/s, where t is calculated in seconds. That said, when we know what’s what by differentiating sin(π²t), we get π²cos(π²t) as an outcome of the chain theory, so we need to take into consideration this additional coefficient when we combine them. You recognize that sin ‘t’ is an antiderivative of cos, so it is rational to anticipate that an antiderivative of cos(π²t) would include sin(π²t). Now moving on to Anie, you want to evaluate Thus, Jessica has ridden 50 ft after 5 sec. Find out who is going to win the horse race?įirst, you need to combine both functions over the interval (0,5) and notice which value is bigger. If Jessica can ride at a pace of f(t)=5+2t ft/sec and Anie can ride at a pace of g(t)=10+cos(π²t) ft/sec.
They are riding the horses through a long, straight track, and whoever reaches the farthest after 5 sec wins a prize. Two jockeys-Jessica and Anie are horse riding on a racing circuit. Using First Fundamental Theorem of Calculus Part 1 Example Since a definite integral has a lower and upper bound, when you take its derivative you just plug in the upper bound into the function inside the integral times the derivate of the upper bound (minus) the lower bound substituted into the function times its d. Lower limit of integration is a constant. \ĭerivative matches the upper limit of integration. The Fundamental Theorem of Calculus denotes that differentiation and integration makes for inverse processes. However, what creates a link between the two of them is the fundamental theorem of calculus (FTC). Both are inter-related to each other, even though the former evokes the tangent problem while the latter from the area problem.
– differential calculus and integral calculus. There are 2 primary subdivisions of calculus i.e. Before proceeding to the fundamental theorem, know its connection with calculus.