Exercises given in lecture on the day in parantheses.

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1 A.Miller M22 Fall 23 Exercises given in lecture on the day in parantheses. The ɛ δ game. lim x a f(x) = L iff Hero has a winning strategy in the following game: Devil plays: ɛ > Hero plays: δ > Devil plays: x such that < x a δ Pay-Off: Hero wins iff f(x) L ɛ. (9-8 Wed) Consider lim x 3 x 2 = 9. (a) Write out the ɛ δ game for this limit. (b) Show that δ = ɛ is not a winning strategy for Hero because if Devil plays ɛ = and Hero plays δ = = then Devil wins with x = 4. (c) More generally for any ɛ > put δ = ɛ. Find a real number x which is a win for the Devil: x 3 δ and x 2 9 > ɛ. 2. (9-8 Wed) Find all pairs of real numbers a, b such that a + b = a + b 3. (9-2 Fri) Find a polynomial q(x) such that x 5 = (x )q(x) 4. (9-2 Fri) Find x 5 lim x x 3 5. (9-23 Mon) Find (a) lim θ sin(sin(θ)) sin(θ) (b) lim θ sin(sin(θ)) θ

2 6. (9-25 Wed) Give a direct proof from the definition of derivative that ( ) d = dx x x 2 The definition of derivative is given on the first page of Chapter 4. An example of a direct proof is Example 2. page 59 where a direct proof is given that d dx (x2 ) = 2x. This problem is the same as 4.5 : (9-25 Wed) Give a direct proof from the definition of derivative that This is the same as 4.5 : 5. d ( ) x = dx 2 x 8. (9-27 Fri) (f g) f g? Let f(x) = x and g(x) = Calculate f, g, and (fg). Verify (f g) f g. Or are they equal? 9. (9-27 Fri) Let f(x) = x x x. and g(x) = x. Is it true that ( ) f f g g. (9-27 Fri) Suppose p(x) and q(x) are polynomials which are not constant. Show that (p q) p q.. (-2 Wed) Find the derivative of sin(sin(sin θ)). 2. (-2 Wed) Find the derivative of sin 3 θ. Recall that sin 3 θ = def (sin θ) 3 3. (-2 Wed) Similarly recall that sin θ = def sin θ which is not the arcsine. The difference is the multiplicative inverse and the inverse with respect to composition of functions. Can the two coincide? 2

3 Find a function f with domain and range the non-zero real numbers such that if g(x) = def for all non-zero x, then f(g(x)) = x = g(f(x)) for all f(x) non-zero x. 4. (-4 Mon) Use the intermediate value theorem to show that has a zero between and 2. f(x) = 2x 2 + x 5 x 5. (-4 Mon) Let f(x) = x cos(x). Use Rolle s Theorem to show that for some a with < a < π 2 we have that f (a) =. 6. (-4 Mon) Suppose f is a twice differentiable function on the whole real line. Suppose f has at least three zeros. Prove that f must have at least one zero. 7. (-4 Mon) Use the Mean-value Theorem to find c with a < c < b such that f f(b) f(a) (c) = b a where f(x) = x and [a, b] = [, 2]. 8. (-4 Mon) Let f(x) = x x. Is f differentiable? If so, find its derivative. Is f twice differentiable? If so, find f. 9. (-8 Fri) Suppose f(θ) = sin(π sin(θ)). Find all critical points, determine all intervals where it is increasing, and determine all intervals where it is decreasing. (Recall that critical points are the same as stationary points.) 2. (-8 Fri) Suppose x 2 if x f(x) = ax 3 + bx 2 + cx + d if < x < x 3 + if x 3

4 Find a, b, c, d so that f is differentiable everwhere. 2. (-23 Wed) Let x = t2 + t 2 y = 2t + t 2 for < t < Show that this parameterizes the unit circle except for the point (, ). Find dx, dy, and dt dt ds dt =def (dx ) 2 + dt ( ) 2 dy dt for each of the points (, ), (, ), and ( 3, 4). 5 5 Trig Problem: If we reparameterize by substituting ( ) θ t = tan 2 then show that x = cos(θ) and y = sin(θ). 22. (-25 Fri) Let x = a + r cos(t) y = b + r sin(t) t 2π (a) Show that this is a parameterization of the circle of radius r center (a, b). (b) Show that the curvature dα ds is constant: r. 23. (-28 Mon) Given < a < b show that there is a c with a < c < b such that b 3 a 3 b 2 a 2 = 3 2 c 24. (-28 Mon) Given π < a < b < π show that there is a c with a < c < b 2 2 such that cos(b) cos(a) sin(a) sin(b) = tan(c) 25. (-28 Mon) Find lim x sin(x) x 4

5 26. (-6 Wed) The Brouwer fixed point theorem for the unit interval says that if f is a continuous map taking [, ] into [, ] then f(x) = x for some x with x. Show that the intermediate value theorem implies the Brouwer fixed point theorem for the unit interval. Hint: consider g(x) = f(x) x. 27. (-6 Wed) Suppose f is continuous on [a, b] and c, c 2,..., c n are in [a, b]. Show that there exists c in [a, b] such that f(c) = f(c ) + f(c 2 ) + + f(c n ) n Hint: show that the average is between the min and the max. 28. (-6 Wed) Find (a) 7 k=3 k(k + ) (b) 5 k= k (c) 3 k= k sin(k π 2 ) 29. (-6 Wed) Suppose n k= (2a k + b k ) = 4 and n k= a k = 2. Then what is n k= 2b k? 3. (-6 Wed) Suppose n k= (2a k + ) = 37. Then what is 9 k= 9a k+? 3. (-6 Wed) Suppose n k= 2a k = 7 and 2n k= 3a k = 7. Then what is 2n k=n+ 5a k? 32. (-8 Fri) Define f(x) = { 3 if x = if x Draw the graph of f. What is I = 2 f(x)dx? Using the ɛ δ definition of limit show that f is Riemann integrable, i.e., given ɛ > what should δ > be? 33. (-8 Fri) Define { if x = f(x) = if x x 5

6 (a) For the partition = x < x < < x 5 = of [, ] < 6 < 8 < 4 < 2 < and choice of right-hand points c k = x k what is 5 f(c k ) x k k= (b) In terms of n what is n k= f(c k) x k for the partition (c) Show that f(x)dx is. 34. (-3 Wed) Given that < 2 n < 2 n 2 < < 4 < 2 < 3 f(x)dx = 2 and 2 3 2f(x)dx = 2 what is 2 3f(x)dx 35. (-3 Wed) Find π f(sin(x)) cos(x)dx where f(u) = ( ) u 3 + e u u u 2 + u (-3 Wed) Let f be the function from exercise 32: { 3 if x = f(x) = if x Show that f has no antiderivative, i.e., there is no F with F (x) = f(x) for every x. 6

7 37. (-3 Wed) Define f by f(x) = { 2x if x if x < (a) Show that f is continuous. (b) Draw the graph of f for x 2. Draw the area under the graph f and above the x-axis for x in [, 2]. (c) Using the area formula for rectangles and triangles show that 2 f(x)dx = 3 (d) Let F (x) be the antiderivative of f: { x F (x) = 2 x if x x if x < (e) Using the Fundamental Theorem of Calculus note that 2 f(x)dx = F (2) F () = (2 2 2) () = 2 (f) Is the Fundamental Theorem wrong? Does 3 = 2? What is this? 38. (-3 Wed) The derivative of F (x) = x is f(x) = x 2 so by the Fundamental Theorem dx = F () F ( ) = ( ) () = 2 x2 But x 2 > so how can the integral be negative? 39. (2-4 Wed) Use the cylindrical shell method to find the volume when the area bounded by y = x 3 + x +, x =, x =, and y = is rotated about the y-axis. What is the volume if the same area is rotated about the x-axis? 4. (2-4 Wed) Find the volume when the area bounded by y = cos(x 2 ), x =, x = π, and y = is rotated about the y-axis. 2 7