Analysis/Calculus Review Day 2

Size: px

Start display at page:

Download "Analysis/Calculus Review Day 2"

Tracey Stevenson
5 years ago
Views:

1 Analysis/Calculus Review Day 2 AJ Friend ajfriend@stanford.edu Arvind Saibaba arvindks@stanford.edu Institute of Computational and Mathematical Engineering Stanford University September 20, 2011

2 Continuity Definition The function f : A B is continuous at x 0 A if, ǫ > 0, δ > 0 such that if x 1 x 0 < δ, then f(x 1 ) f(x 0 ) < ǫ. Definition The function f : A B is continuous on A if it is continuous at every point of A. This equivalent to saying that for every convergent sequence x k x 0, we have f(x k ) f(x 0 ).

3 Examples f(x) = x 2 { 0, if x < 0 f(n) = 1, if x 0 { 0, if x Q f(n) = 1, if x Q

4 Examples Let f : R R be the identity function x x. Show that f is continuous. Solution Fix x 0 R. By definition we must find δ > 0 for given ǫ > 0 such that x x 0 < δ implies f(x) f(x 0 ) < ǫ. This is trivially true if one chooses δ = ǫ. Hence f is continuous.

5 Theorem If x n x 0 then f : A B is continuous at x 0 iff f(x n ) f(x 0 ). Proof. One direction: Given ǫ > 0, δ > 0 such that if y x 0 < δ, then f(y) f(x 0 ) < ǫ. Choose N large enough so that x n x 0 < δ n N. Then f(x n ) f(x 0 ) < ǫ n N.

6 Properties of Continuous Functions f +g, αf, f g, f/g (if g 0) are continuous Composition f(g(x)) is continuous If A is compact then f(a) is also compact If f : A R, A compact, f continuous, then f achieves its sup and inf. Counterexample: 1/x on (0, 1).

7 Uniform Continuity Definition A function f : A B is uniformly continuous if ǫ > 0, δ > 0 such that if x y < δ then f(x) f(y) < ǫ. Note: does not depend on the point of continuity. Examples: f(x) = x 2 not uniformly continuous sin(x) is uniformly continuous

8 Lipschitz functions Definition A function f : A B is Lipschitz continuous if K > 0 such that f(x) f(y) K x y. Properties: Bounds the rate of change of the function Lipschitz functions are uniformly continuous (proof?) functions with bounded derivative are Lipschitz Example: x defined on [0,1] is not Lipschitz

9 Sequences of functions and Convergence Definition (Pointwise Convergence) Given a sequence of functions of a set E, namely f n : E R. Suppose that for all x E lim n f n(x) = f(x) then we say that {f n } converges pointwise to f. The disadvantage with pointwise convergence is even if f n are all continuous, f need not be continuous. For example, consider { 0 x 1 f n (x) = k kx +1 0 x < 1 k In this case, for each x [0,1], f k (x) converges. If x 0,f k (x) 0 (since f k (x) = 0 for large k), while if x = 0,f k (x) = 1. The limit is thus { 0 x 0 f(x) = 1 x = 0

10 Uniform Convergence Definition We say {f n } converges uniformly to f (denoted f n f(uniformly), if for all ǫ > 0, there exists an N such that for all n N f n (x) f(x) ǫ, x For example, on R consider the following sequence, { 0 x < k f k (x) = 1 x k Clearly, f k (x) 0 pointwise. (for k large f k (x) = 0). However, f k does not converge to 0 uniformly, because there are points x such that f(x) 0, is not small no matter how large k is.

11 contd. Example Let f n (x) = sinx n,f n : R R Show that f n 0 uniformly as n. Solution We must show that f n (x) 0 = f n (x) gets small independent of x as n. But f n (x) = sinx /n 1/n which gets small independent of x as n. Things to note Uniform convergence implies Pointwise convergence but not vice versa. If {f n } is a sequence of continuous functions and f n f(uniformly), then f is a continuous function.

12 Theorem If f n : X B are continuous and f n f uniformly, then f is continuous. Proof. Given ǫ > 0, f n f uniformly, so N such that We have f N (x) f(x) ǫ/3 x X. f(x) f(a) f(x) f N (x) + f N (x) f N (a) + f N (a) f(a) 2/3ǫ+ f N (x) f N (a). f N is continuous, so let δ > 0 be such that if x a < δ then f N (x) f N (a) ǫ/3, which would give f(x) f(a) ǫ.

13 Differentiation For a function f : [a,b] R, f is called differentiable at x 0 (a,b) if the limit f f(x 0 +h) f(x 0 ) (x 0 ) = lim h 0 h exists. One also writes df/dx for f (x).

14 Tangents

15 Properties of Derivatives Let f,g be differentiable on [a,b]. Then, (cf) (x) = cf (x) (f +g) (x) = f (x)+g (x) Product Rule. Quotient Rule Chain Rule (f g) (x) = f (x)g(x)+f(x)g (x) (f/g) (x) = f (x) g(x) g (x)f(x) g 2 (x) h(x) = g(f(x)) h (x) = g (f(x))f (x)

16 Theorem If f : (a,b) R is differentiable at c (a,b) and f has a maximum (respectively minimum) at c, then f (c) = 0. Proof. Let f have a maximum at c. Then for h 0, [f(c +h) f(c)]/h 0 So, letting h 0,h 0 we get f (c) 0. Similarly, for h 0 we obtain f (c) 0. Hence, f (c) = 0.

17 Rolle s Theorem Theorem If f : [a,b] R is continuous, f is differentiable on (a, b) and f(a) = f(b) = 0 then there is a number c (a,b) such that f (c) = 0.

18 Proof of Rolle s theorem Theorem If f : [a,b] R is continuous, f is differentiable on (a,b) and f(a) = f(b) = 0 then there is a number c (a,b) such that f (c) = 0. Proof. If f(x) = 0 for all x [a,b] we can choose any c and we are done. So assume that f is not identically zero. From the min-max theorem we know that there is a point c 1 where f assumes its maximum and a point c 2 where f assumes its minimum. By our assumption and the fact that f(a) = f(b) = 0 at least one of c 1,c 2 (a,b). If c 1 (a,b) we have f (c 1 ) = 0; similarly for c 2.

19 Mean Value Theorem Theorem If f : [a,b] R is continuous, f is differentiable on (a,b), there is a point c (a,b) such that f(b) f(a) = f (c)(b a) Corollary If f is differentiable on (a,b) and f (x) = 0 for all x (a,b) then f is constant on (a,b).

20 MVT Proof Apply Rolle s theorem to φ(x) = f(x) f(a) (x a) f(b) f(a) b a

21 L Hopital s Rule Suppose f and g are differentiable on ]a,b[ and g (x) 0 for x ]a,b[. Suppose f (x) g (x) A as x a If f(x) 0 and g(x) 0 as x 0, or if g(x) as x a, then Example sinx lim x 0 x f(x) g(x) A as x a d/dx(sinx) cosx = lim = lim x 0 d/dx(x) x 0 1 = 1

22 Properties of Integration For integrable functions f 1,f 2,f f 1 +f 2 is integrable and cf is integrable. (Linearity) If f 1 f 2, Let a b c. Then, c a b a f(x)dx = Suppose sup f(x) M, then f 1 (x)dx b a b a f 2 (x)dx c f(x)dx + f(x)dx b f 1 f 2 is integerable. If f(x) is integrable b a f(x)dx M(b a) b f(x)dx b a a f(x) dx

23 Fundamental Theorem of Calculus If f is integrable over [a,b], define F(x) = x a f(t)dt Then, F is continuous. If f is continuous at x 0 (a,b) then F is differentiable at x 0 and F (x 0 ) = f(x 0 ). F is called the anti-derivative of f. Theorem Let f : [a,b] R be continuous. Then f has an antiderivative F and b a f(x)dx = F(b) F(a) If G is any other antiderivative of f, then we also have that f(x)dx = G(b) G(a). b a

24 Integration by parts The product rule for differentiation is (f g) (x) = f (x)g(x)+f(x)g (x) Thus, we can write f(x)g (x)dx = f(x)g(x) f (x)g(x) Example Integrate log xdx We use the above formula with f(x) = logx and g (x) = 1. Therefore, 1 logxdx = (logx)x x xdx = x logx x

25 Differentiation under the Integral sign Theorem (Leibniz Rule) If f is continuous on [a,b] and u(x),v(x) are differentiable functions of x whose values lie in [a,b], then d dx v(x) u(x) f(t)dt = f(v(x))v (x) f(u(x))u (x)

Analysis/Calculus Review Day 2

Analysis/Calculus Review Day 2 Arvind Saibaba arvindks@stanford.edu Institute of Computational and Mathematical Engineering Stanford University September 14, 2010 Limit Definition Let A R, f : A R and