Presentation on theme: "Aim: What is the Fundamental Theorem of Calculus?"— Presentation transcript:
1 Aim: What is the Fundamental Theorem of Calculus? Do Now:-1/18
2 Connection: Differentiation & Integration Two branches of calculus:Differentiation - rate of changeIntegration – accretion (area)Inverses of each otherΔyΔxsecantΔyΔxarea of rectangletangentarea of regionprecalculusprecalculus
3 Connection: Differentiation & Integration Calculus is the study of limitsderivative of a functiontwo most important limitsdefinite integralFundamental Theorem of Arithmeticwhole numbers can be factored into product of primesFundamental Theorem of Algebranth degree polynomial has n roots
4 The Fundamental Theorem of Calculus If a function of f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], thenGuidelinesYou now have a way to evaluate a definite integral without using the limit of a sum.Use the following notationIt is not necessary to use the constant of integration C
7 Finding Area of RegionFind the area of the region bounded by the graph of y = 2x2 – 3x + 2, the x-axis, and the vertical lines x = 0 and x = 2.Integrate [0, 2]Find F(x)Evaluate F(2) – F(0)
8 Mean Value Theorem of Integrals If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such thatcf(c)f= average value of f on [a, b]absomewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is equal to the area of the region under the curve.
9 Average Value of a Function If f is integrable on the closed interval [a, b], then the average value of f on the interval isFind the average value of f(x) = 3x2 – 2x on the interval [1, 4]
10 Average Value of a Function Find the average value of f(x) = 3x2 – 2x on the interval [1, 4]3x2 – 2x = 16c = 8/3cf(c)average value
11 Model ProblemAt different altitudes in earth’s atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled bywhere x is the altitude in kilometers. What is the average speed of sound over the interval [0, 80]?
13 Accumulation Function Definite Integral as a NumberDefinite Integral as a Function of xConstantF is a function of xnew variable of integrationf is a function of xf is a function of tConstantConstantAccumulation function: area accumulates under a curve from fixed value of (t = a) to a variable value (t = x)Note: definite integral is not function of variable of integration
15 2nd Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then, for every x in the interval,EvaluateCaution: if the upper limit is a function of x, ex. x2, then the answer is multiplied by the derivative of the upper limit term.
16 Model ProblemFind the derivative of F(x)chain ruleverification