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

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(x 1 ) = y 1 substitute the values into the function and solve for c: y = F(x) + C  y 1 = F(x 1 ) + C C = y 1 - F(x 1 )

9. Example: Solve, dr/dp = 3 sin p with 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) =  a x f(t)dt is an antiderivative of f, and equivalently:

11. Example:

12.  That’s allFolks! Have a good Summer! God Bless