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Antiderivatives and Indefinite Integration

Antiderivatives and Indefinite Integration. Lesson 5.1. Reversing Differentiation. An antiderivative of function f is a function F which satisfies F’ = f Consider the following: We note that two antiderivatives of the same function differ by a constant.

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Antiderivatives and Indefinite Integration

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  1. Antiderivatives and Indefinite Integration Lesson 5.1

  2. Reversing Differentiation • An antiderivative of function f is • a function F • which satisfies F’ = f • Consider the following: • We note that two antiderivatives of the same function differ by a constant

  3. Reversing Differentiation • General antiderivativesf(x) = 6x2 F(x) = 2x3 + C • because F’(x) = 6x2 k(x) = sec2(x) K(x) = tan(x) + C • because K’(x) = k(x)

  4. Differential Equation • A differential equation in x and y involves • x, y, and derivatives of y • Examples • Solution – find a function whose derivative is the differential given

  5. Differential Equation • When • Then one such function is • The general solution is

  6. "The antiderivative of f with respect to x" Notation for Antiderivatives • We are starting with • Change to differential form • Then the notation for antiderivatives is

  7. Basic Integration Rules • Note the inverse nature of integration and differentiation • Note basic rules, pg 286

  8. Practice • Try these

  9. Finding a Particular Solution • Given • Find the specific equation from the family of antiderivatives, whichcontains the point (3,2) • Hint: find the general antiderivative, use the given point to find the value for C

  10. Assignment A • Lesson 5.1 A • Page 291 • Exercises 1 – 55 odd

  11. Slope Fields • Slope of a function f(x) • at a point a • given by f ‘(a) • Suppose we know f ‘(x) • substitute different values for a • draw short slope lines for successive values of y • Example

  12. Slope Fields • For a large portion of the graph, when • We can trace the line for a specific F(x) • specifically when the C = -3

  13. Finding an Antiderivative Using a Slope Field • Given • We can trace the version of the original F(x) which goes through the origin.

  14. Vertical Motion • Consider the fact that the acceleration due to gravity a(t) = -32 fps2 • Then v(t) = -32t + v0 • Also s(t) = -16t2 + v0t + s0 • A balloon, rising vertically with velocity = 8 releases a sandbag at the instant it is 64 feet above the ground • How long until the sandbag hits the ground • What is its velocity when this happens? Why?

  15. Rectilinear Motion • A particle, initially at rest, moves along the x-axis at acceleration • At time t = 0, its position is x = 3 • Find the velocity and position functions for the particle • Find all values of t for which the particle is at rest Note Spreadsheet Example

  16. Assignment B • Lesson 5.1 B • Page 292 • Exercises 57 – 93, EOO

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