Continuity and Differentiability in Calculus

1. Chapter 2 Calculus: Hughes-Hallett The Derivative

2. Continuity of y = f(x) • A function is said to be continuous if there are no “breaks” in its graph. • A 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.

3. Continuous Functions- • The function f is continuous at x = c if f is defined at x = c and • The function is continuous on an interval [a,b] if it is continuous at everypoint in the interval. • If 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.)

4. Definition of Limit- • Suppose a function f, is defined on an interval around c, except perhaps not at the point x = c. • The 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). • In Symbols:

5. Properties of Limits- • Assuming all the limits on the right hand side exist:

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

7. 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.

8. The Derivative is -- • Physically- an instantaneous rate of change. • Geometrically- the slopeof thetangent lineto thegraph of the curve of the function at a point. • Algebraically- the limit of the difference quotient as h 0 (if that exists!).In symbols:

9. First Derivative Interpretation- • If f’ > 0 on an interval, then f is increasing over the interval. • If f’ < 0 on an interval, then f is decreas-ingover the interval.

10. 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:

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

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

13. Continuous Functions- • The function f is continuous at x = c if f is defined at x = c and • The function is continuous on an interval [a,b] if it is continuous at everypoint in the interval. • If 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.)

14. Continuity of y = f(x) • A function is said to be continuous if there are no “breaks” in its graph. • A 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.

15. Theorem on Continuity- • Suppose that f and g are continuous on an interval and that b is a constant. Then, on that same interval: • 1. bf(x) is continuous. • 2. f(x) + g(x) is continuous. • 3. f(x)g(x) is continuous. • 4. f(x)/g(x) is continuous, provided ` on the interval.

16. Differentiability- • A function f is said to be differentiable at x = a if f’(a) exists. • Theorem: If f(x) is differentiable at a point x = a, then f(x) is continuous at x = a.

17. Linear Tangent Line Approximation- • Suppose f is differentiable at x = a. Then, for values of x near a, the tangent line approximation to f(x) is: • The 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