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Chapter 6: Calculus~ Hughes-Hallett

Chapter 6: Calculus~ Hughes-Hallett. Constructing the Antiderivative Solving (Simple) Differential Equations The Fundamental Theorem of Calculus (Part 2). Review: The Definite Integral. Physically - is a summing up Geometrically - is an area under a curve

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Chapter 6: Calculus~ Hughes-Hallett

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  1. Chapter 6: Calculus~Hughes-Hallett • Constructing the Antiderivative • Solving (Simple) Differential Equations • The Fundamental Theorem of Calculus (Part 2)

  2. Review: The Definite Integral • Physically - is a summing up • Geometrically - is an area under a curve • Algebraically - is the limit of the sum of the rectangles as the number increases to infinity and the widths decrease to zero:

  3. Review of The Fundamental Theorem of Calculus (Part 1) If f is continuous on the interval [a,b] and f(t) = F’(t), then: • In words: the definite integral of a rate of change gives the total change.

  4. Properties of Antiderivative: 1. [f(x)  g(x)]dx = f(x)dx  g(x)dx (The antiderivative of a sum is the sum of the antiderivatives.) 2. cf(x)dx = cf(x)dx (The antiderivative of a constant times a function is the constant times the anti- derivative of the function.)

  5. The Definition of Differentials (given y = f(x)) 1. The Independent Differential dx: If x is the independent variable, then the change in x, x is dx; i.e. x = dx. 2. The Dependent Differential dy: If y is the dependent variable then: i.) dy = f ‘(x) dx, if dx  0 (dy is the derivative of the function times dx.) ii.) dy = 0, if dx = 0.

  6. Using the differential with the antiderivative.

  7. Solving First Order Ordinary Linear Differential Equations • To solve a differential equation of the form dy/dx = f(x) write the equation in differential form: dy = f(x) dx and integrate: dy = f(x)dx y = F(x) + C, given F’(x) = f(x) • If initial conditions are given y(x1) = y1 substi-tute the values into the function and solve for c: y = F(x) + C  y1 = F(x1) + C C = y1 - F(x1)

  8. Example: Solve, dr/dp = 3 sin pwith r(0)= 6, i.e. r= 6 when p = 0 • Solution:

  9. The Fundamental Theorem of Calculus (Part 2) If f is a continuous function on an interval, & if a is any number in that interval, then the function F, defined by F(x) =  ax f(t)dt is an antiderivative of f, and equivalently:

  10. Example:

  11. That’s all Folks! Have a good Christmas! God Bless

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