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The First Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus. First Fundamental Theorem. Take the Antiderivative. Evaluate the Antiderivative at the Upper Bound. Evaluate the Antiderivative at the Lower Bound. Subtract the Lower Bound Value from the Upper Bound Value.

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The First Fundamental Theorem of Calculus

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  1. The First Fundamental Theorem of Calculus

  2. First Fundamental Theorem • Take the Antiderivative. • Evaluate the Antiderivative at the Upper Bound. • Evaluate the Antiderivative at the Lower Bound. • Subtract the Lower Bound Value from the Upper Bound Value.

  3. The Definite Integral If f is a continuous function, the definite integralof f from a to b is defined to be The function f is called the integrand, the numbers a and b are called the limits of integration, and the variable x is called the variable of integration.

  4. The Definite Integral is read “the integral, from a to b of f(x)dx.” Also note that the variable x is a “dummy variable.”

  5. The Definite Integral As a Total If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

  6. Net Change By the Fundamental Theorem of Calculus we have, for an indefinite integral function F of f: This can be rewritten as follows Definition The quantity F(b) – F(a) is the net change of the function F over the interval [a,b]. The derivative F’(x) is the rate of change of the function F.

  7. So now we can do all kinds of summing problems before we even mention an antiderivative. Historically, that’s what scientists had to do before calculus. Here’s why it mattered to them:

  8. The Definite Integral As a Total Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

  9. The calculus pioneers knew that the area would still yield distance, but what was the connection to tangent lines? And was there an easy way to find these irregularly-shaped areas? The Definite Integral

  10. Area Under a Graph Width: (n rect.) a b Idea: To find the exact area under the graph of a function. Method: Use an infinite number of rectangles of equal width and compute their area with a limit.

  11. Geometric Interpretation (All Functions) R1 R3 a b R2 Area of R1 – Area of R2 + Area of R3

  12. Area Using Geometry Ex. Use geometry to compute the integral Area =4 Area = 2

  13. Computing Area Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x3 is nonnegative on [0, 2]. Antiderivative Fund. Thm. of Calculus

  14. Evaluate:

  15. Evaluate:

  16. Evaluate Definite Integralswith your calculator. • Enter the function in y =. • GRAPH & CALC (2nd TRACE)  Option 7 or MATH  Option 9 fnInt(y1,x, __, __) • Enter Lower Bound • Enter Upper Bound

  17. AREA under f (x)down to the x-axisfrom x = a to x = b Use your graphing calculator to graph the integrand and determine whether the integral is… positive, negative, or zero.

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