2. Chapter 2 Calculus: Hughes-Hallett The Derivative

3. Continuity of y = f(x) zA function is said to be continuous if there are no “breaks” in its graph. zA function is continuous at a point x = a if the value of f(x)  L, a number, as x  a for values of x either greater or less than a.

4. Continuous Functions- zThe function f is continuous at x = c if f is defined at x = c and zThe function is continuous on an interval [a,b] if it is continuous at everypoint in the interval. zIf f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval. (A theorem.)

5. Definition of Limit- zSuppose a function f, is defined on an interval around c, except perhaps not at the point x = c. zThe limit of f(x) as x approaches c: is the number L (if it exists) such that f(x) is as close to L as we please whenever x is suffici- ently close to c (but x c). zIn Symbols:

6. Properties of Limits- zAssuming all the limits on the right hand side exist:

7. Limits at Infinity- zIzIf f(x) gets as close to a number L as we please when x gets sufficiently large, then we write: zSzSimilarly, if f(x) approaches L as x gets more and more negative, then we write:

8. Average Rate of Change- The average rate of change is the slope of the secant line to two points on the graph of the function.

9. The Derivative is -- zPhysically- an instantaneous rate of change. zGeometrically- the slope of the tangent line to the graph of the curve of the function at a point. zAlgebraically- the limit of the difference quotient as h  0 (if that exists!). In symbols:

10. First Derivative Interpretation- zIf f’ > 0 on an interval, then f is increasing over the interval. zIf f’ < 0 on an interval, then f is decreas- ing over the interval.

11. Derivative Symbols: If y = f(x) = then each of the following symbols have the same meaning: And at a particular point, say x = 2, these symbols are used:

12. Basic Formulas (1): zDerivative of a constant: If f(x) = k, the f’(x) = 0, k - a constant zDerivative of a linear function: If f(x) = mx + b, then f’(x) = m zDerivative of x to a power:

13. Second Derivative Interpretation- zIf f’’ > 0 on an interval, then f’ is increasing, so the graph of f is concave up there. zIf f’’ < 0 on an interval, then f’ is decreasing, so the graph of f is concave down there. zIf y = s(t) is the position of an object at time t, then: zVelocity: v(t) = dy/dt = s’(t) = zAcceleration: a(t) =

14. Continuous Functions- zThe function f is continuous at x = c if f is defined at x = c and zThe function is continuous on an interval [a,b] if it is continuous at everypoint in the interval. zIf f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval. (A theorem.)

15. Continuity of y = f(x) zA function is said to be continuous if there are no “breaks” in its graph. zA function is continuous at a point x = a if the value of f(x)  L, a number, as x  a for values of x either greater or less than a.

16. Theorem on Continuity- zSuppose that f and g are continuous on an interval and that b is a constant. Then, on that same interval: z1. bf(x) is continuous. z2. f(x) + g(x) is continuous. z3. f(x)g(x) is continuous. z4. f(x)/g(x) is continuous, provided ` on the interval.

17. Differentiability- zA function f is said to be differentiable at x = a if f’(a) exists. zTheorem: If f(x) is differentiable at a point x = a, then f(x) is continuous at x = a.

18. Linear Tangent Line Approximation- zSuppose f is differentiable at x = a. Then, for values of x near a, the tangent line approximation to f(x) is: zThe expression is called the local linearization of f near x = a. We are thinking of a as fixed, so that f(a) and f’(a) are constant. The error E(x), is defined by and