Aim: What is the Fundamental Theorem of Calculus? Do Now: -1/18

Connection: Differentiation & Integration Two branches of calculus: Differentiation - rate of change Integration – accretion (area) Inverses of each other Δy Δx secant Δy Δx area of rectangle tangent area of region precalculus precalculus

Connection: Differentiation & Integration Calculus is the study of limits derivative of a function two most important limits definite integral Fundamental Theorem of Arithmetic whole numbers can be factored into product of primes Fundamental Theorem of Algebra nth degree polynomial has n roots

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], then Guidelines You now have a way to evaluate a definite integral without using the limit of a sum. Use the following notation It is not necessary to use the constant of integration C

Evaluating a Definite Integral

Evaluating an Absolute Value

Finding Area of Region Find 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)

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 that c f(c) f = average value of f on [a, b] a b somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is equal to the area of the region under the curve.

Average Value of a Function If f is integrable on the closed interval [a, b], then the average value of f on the interval is Find the average value of f(x) = 3x2 – 2x on the interval [1, 4]

Average Value of a Function Find the average value of f(x) = 3x2 – 2x on the interval [1, 4] 3x2 – 2x = 16 c = 8/3 c f(c) average value

Model Problem At different altitudes in earth’s atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled by where x is the altitude in kilometers. What is the average speed of sound over the interval [0, 80]?

Model Problem Sum of 5 integrals -

Accumulation Function Definite Integral as a Number Definite Integral as a Function of x Constant F is a function of x new variable of integration f is a function of x f is a function of t Constant Constant Accumulation 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

Model Problem Evaluate option 1 option 2

2nd Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then, for every x in the interval, Evaluate Caution: if the upper limit is a function of x, ex. x2, then the answer is multiplied by the derivative of the upper limit term.

Model Problem Find the derivative of F(x) chain rule verification

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