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Chapter 6: Calculus~ Hughes-Hallett

This chapter focuses on constructing the antiderivative, solving simple differential equations, and understanding the fundamental theorem of calculus. It also reviews the definite integral and its physical, geometric, and algebraic interpretations.

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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. Differential and Integral Formulas

  5. 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 antiderivative of the function.)

  6. 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.

  7. Using the differential with the antiderivative.

  8. 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 substitute the values into the function and solve for c: y = F(x) + C  y1 = F(x1) + C C = y1 - F(x1)

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

  10. 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:

  11. Example:

  12. That’s all Folks! Have a good Summer! God Bless

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