Chapter 1 Summary

1. Chapter 1 Summary Math 1231: Single-Variable Calculus

2. Roots or Zeros: Definition • Given a function f(x), we call x = a is a ROOT or ZERO of f(x) if f(a) = 0. • Given a equation f(x)=0, we call x = a is a root or zero or SOLUTION of the equation if f(a) = 0. Example: x = π/2 is a root of cos(x); Example: x = 1 is a root of x2 + x = x + 1.

3. Limit • limx->af(x) = L exists if, for ANY sequence of x values that approach a, f(x) always becomes sufficiently close to L. • Example: limx->2 x2 = 4 means, for any sequence of x values, • x = 2.1, 2.01, 2.001, 2.0001, … • x = 1.9, 1.99, 1.999, 1.9999, … • x = 2.2, 1.8, 2.02, 1.98, 2.002, 1.998, … x2 always sufficiently approaches 4.

4. When Does Limit Not Exist? • If f(x) approaches two different values when x approaches a along two different sequences, then f(x) does not have a limit. • Example: f(x) = sin(1/x) does not have a limit when x approaches 0, because by taking x to be • x = 1/π, 1/2π, 1/3π, …, f(x) 0 • x = 1/0.5π, 1/2.5π, 1/4.5π, …, f(x)1

5. More Examples For any point x = a, limx-> af(x) does not exist. So f(x) is discontinuous everywhere. limx-> af(x) does not exist for all the points except x = 0. Furthermore, f(x) is continuous only at x = 0.

6. Continuity f(x) = f1(x) f(x) = f2(x) f(x) = f3(x) 2 4

7. Continuity f(x)iscontinuousataif limx->af(x)=f(a) Continuityimplies we can switch the order of limandf Example: Suppose that limx->1 g(x) = π, find limx->1 cos(g(x)).

8. Miscellaneous • sine and cosine functions are bounded by -1 and 1.

9. Miscellaneous • I.V.T. is only valid when f(x) is continuous.