Math 44 Activity (Due by end of class July 3) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through y equations y m 3 make this true.} 3,, y m 3, to be tangent to y, the system of must have eactly one solution. Find the values of m that. Find the center of the circle that passes through the points,7, 8,6, and 7, {Hint: The center must lie on the perpendicular bisectors of the segments through the given points.}.
3. Prove that if the graphs of the linear functions f m and g then mn. n are perpendicular {Hint: Apply the Pythagorean Theorem to the following triangle.} y m m, m m n n ; y n 4. If f and g 3 ; 3, then find all values of so that f g. {Hint: Set the two formulas for f equal to g, and solve carefully.} 3 6n n 3n 67 5. a) Use polynomial long division to find the largest integer n, so that is an n 3 integer. 3n 5n 6 3 n 3 6n n 3n 67 6n 9n 3 n 3n 67 n 5 n 67 n 8 49 3 6n n 3n 67 49 So 3n 5n 6. Find the largest integer value of n so that n3 n 3 n 3 is a factor of 49. 4 3 n 3n n 3n 7 b) Use polynomial long division to find the largest integer n, so that is n 3 an integer., n n
6. If f is a real-valued function with the property that f y f f y for all real f 3, then find f {Hint: f f numbers and y, and if a).} b) f 3 {Hint: f 3 f.} c) f 4 {Hint: f 4 f d) f,,.} e) f {Hint: f f f f.} f) f {Hint: f f f f. } g) f {Hint: f f f f. } 7. If f is a real-valued function with the property that f y f f y for all real numbers and y, and if f 3, then find a) f {Hint: f f.} b) f 3 {Hint: f 3 f.} c) f 4 {Hint: f 4 f d) f,.} e) Show that f for all real numbers,. {Hint: f f f f) f {Hint: f f f f.}.} g) f {Hint: f f f f. } f 8. Given that f and f 3 for all, f a) Find f 4 b) Find f 7 c) Find f d) Find f 3 e) Find f 8
Graphing, Limits and Continuity Problems: Use the graphs of the functions f and g to answer problems 9 and. Graph of f Graph of g f g 9. a) f g 3 b) f g c) f g d) 3 f g e) f) f g 3 g) g f h) g g i) f g g j) Solve f g. k) Solve f g. l) Solve f g.. a) lim f g b) lim f g c) lim f g d) f g e) f g g) lim f g h) lim f g lim f) Is f g continuous at? i) f g lim j) lim f g k) lim f g l) lim f g m) f n) f g g lim o) lim f g lim p) lim f g 3
3. Show that the equation 5 4,4. {Hint: Find three sign changes in the interval 4,4, and use the Intermediate Value Theorem. How many real zeros can a third degree polynomial have?}. Find values of a, b, and c that will make f has eactly three solutions in the interval, a b c, continuous. 3, 3 c 6, 3 {Hint: The function will be continuous if its graph is connected. The separate pieces are connected, so just make sure that these pieces join at the values,,3 by solving a b c the system of equations 4a b c 6.} 9 3c 6 3. If lim f g, f g a a) lim f a 4. Let f. a) Find lim f lim, then find a b) lim g c) lim f g a a {Hint: f f g f g, g f g f g. } b) Find lim f c) Find lim f 5. Let f 7. a a) For what value(s) of a, does lim f a {Hint: ; equal a number?.} b) For what value(s) of a, does lim f a? c) For what value(s) of a, does lim f a? {Hint: 7 3 4.}
Derivative Problems: f has no horizontal tangent lines. What is the smallest slope of a tangent line to the graph of this function? {Hint: Check out the derivative.} 6. Show that the function 5 3 7. Find constants a, b, c, and d so that the graph of f a b c d has horizontal tangents at, and,. {Hint: Make the derivative and function values correct.} 8. Here s the graph of the function f on the interval 3,4 3, lim f. 3. It has a vertical asymptote at a) What are the critical numbers of f? b) What is the absolute maimum of f on 3,4? c) What is the absolute minimum of f on 3,4? d) Where does f have local maima? e) Where does f have local minima? f) Find the maimum of f on,. g) Find the minimum of f on,3. h) Find the maimum of f on,. i) Find the minimum of f on,4. j) Find the minimum of f on 3,. k) Where is the function increasing? l) Where is the function decreasing?
9. Given the graph of the derivative of f, answer the following questions. a) Where is f increasing? b) Where is f decreasing? c) Where does f have local maima? d) Where does f have local minima? f e) Which is larger f or f? f) Which is larger f or f? g) Does f have an inflection point at? h) Does f have an inflection point at? i) Which is larger, f or f? {Hint: f f f d.} 3, then find f n n n. If f. {Hint: Simplify the sum first.}. Find f if it is known that f d d. f using the Chain Rule, then replace with.} {Hint: Differentiate
. Here is the graph of the derivative, f, for a function f which is continuous on the interval,8. a) Find the intervals where f is increasing. b) Find the intervals where f is decreasing. c) Find the intervals where f is concave-up. d) Find the intervals where f is concave-down. e) Find the local maima. f) Find the local minima. g) Find the inflection points. 3. Using the graphs of the functions f and g, evaluate the following: Graph of f Graph of g a) f g b) f g c) g f d) f f 3 e) g g f) 4 g f g) f g g 5 h) f g f i) g g g 4
4. Find equations for all the lines through the origin that are tangent to the parabola y 4. 5. If f, then find the derivative of a) f when. b) 6. Let f cosk {Hint: At such a point, f when. c) 4.} f when 4. for k. a) There is a number A so that if k A, then f has no inflection points, and if k A, then f has infinitely many inflection points. Find A. k k cos k k.} {Hint: 4 f k k
b) Show that f has only finitely many local etrema no matter what the value of k. k {Hint: k k sink k, so what can you say about f if or k?} f k y k Rolle s and Mean Value Theorem Problems: f 3 a has two distinct zeros in the interval,? {Hint: If there is such a value of a, then Rolle s Theorem implies that the derivative must equal zero in the interval,.} 7. Is there a value of a so that 3 7 5 3 8. Show that the function f has eactly one real zero. {Hint: It has no positive zeros. Find a sign change, and use the Intermediate Value Theorem to conclude that it has at least one negative zero. Suppose it has two or more, and use Rolle s Theorem as in the previous hint.} 9. Suppose that f eists on ab,, f a f b, but f on, f on ab,. {Hint: Suppose that f c for some c in the interval, in the previous hints.} ab. Show that ab, and use Rolle s Theorem as
f is increasing on,. f f f f {Hint: g and the Mean Value Theorem implies that f f f fc for some c in,. But this means that fc for some c in,. Take it from here.} 3. Suppose that f and f is increasing. Show that the function g 3. Is there a differentiable function f with f 7, f 3, and f on,3 the answer is yes, then give an eample of such a function. If no is the answer, eplain why such a function is impossible. {Hint: Mean Value Theorem}? If Approimating Solutions by Iteration Problems: 3. Attempt to solve the following equations using a cobweb diagram with the given starting value: a) 6, 3 b) 8, 3 lim n n lim n n
c) 5, 6 d) 6, 5 lim n n lim n n e), f) 4 4, 4 lim n n lim n n {Check out the Successive Approimation Worksheet link on the course webpage!}
Integration and Fundamental Theorem Problems: 33. Let f be the function graphed below on the interval [,3]. Note: The graph of f consists of two line segments and two quarter-circles of radius 3. a) Evaluate f d b) Evaluate f d c) Evaluate f 5 8 d 8 d) Evaluate f d e) Evaluate f d f)evaluate f g) If F f t 3 8 dt, then construct the sign chart for the derivative of F on the interval 3 d,3. d.} d a {Hint: Use the Fundamental Theorem of Calculus: f tdt f h) Find the local etrema and absolute etrema for F on the interval,3. i) If H Ft dt, then construct the sign chart for the derivative of H on the interval,3.
f f? 34. Is there a differentiable function f so that f and sin {Hint: For, f t f t dt f t f f and so integrate, plug in, and see what happens.} f t f t dt sint dt, 35. Use the fact that ; 4 4 7 d. 4 to find good upper and lower bounds for b b b g d f d h d.} {Hint: If g f h for a b, then 36. Find all continuous functions f on, so that f tdt a a a f t dt for all in,. {Hint: Use the Fundamental Theorem of Calculus: f tdt f d.} d a 37. If for, tf t dt 5 5, then find the value of 4 f 5. d and the Chain d a {Hint: Use the Fundamental Theorem of Calculus: f tdt f Rule.} 38. If f and g are continuous functions, then what is the value of f f g g d? {Hint: For F f f, notice that F f f F even function. Try the same thing with G g g.}, so F is an
4 39. Suppose that f is continuous and f d, f d, and 4 f d 7. a) Find the value of f d. b) Find the value of f d. c) Find the value of f d) Eplain why there must be an in, with f. d. {Hint: Suppose that f for in,, then what can you say about f e) Eplain why there must be an in, 4 with f 3.5. d?} {Hint: Suppose that f 3.5 for in,4, then what can you say about f 4 d?} 4. Suppose that f and g are continuously differentiable functions with f g for all. Show that there must be a number a so that f a ga. {Hint: Suppose the opposite is true: f g for all, then for, f tdt gtdt. This implies that f g g f happens if g f?}. What 4. Show that the function t t, for 4 4 t t f dt dt, is a constant function. {Hint: What s the derivative of this function? Use the Fundamental Theorem of Calculus and the Chain Rule.}
4 4. a) If f is continuous and f d, then find f 9 b) If f is continuous and d. f d 4, then find f d. 3 {Hint: Make a substitution.} {Hint: Make a substitution.} 5 c) If f is continuous and f d, then find 3 f d. {Hint: Make a substitution.} d) If f is continuous and f d 3, then find f d. {Hint: Make a substitution.} e) If f is continuous and f d 3, then find f 3 d. 3 {Hint: Make a substitution.} 43. The graph of the derivative, f, is given. Complete the table of values for f that f. given 3 4 5 6 f.} {Hint: f f tdt f
b 44. If f d 3a b, then find the value of a 45. If 3 u f t e du dt t a f d., then find the value of b f. {Hint: The Fundamental Theorem of Calculus twice} 46. If f is continuous on,4, f d and f 4, then find 4 47. Suppose that f is a continuous function on, with f be an in {Hint: Suppose that with, with f. f for all in f f. {Hint: Let u in the integral.} d. Show that there must, and arrive at a contradiction. Do the same. Or apply Rolle s Theorem to the function F f t tdt on the interval,.} Without assuming continuity, there doesn t have to be an in Consider f ;. ;, with f. f d From the graph, you can see that there is no in, with f.
48. Find the points on the hyperbola y that are closest to the point,., y, {Hint: The slope from, to the closest points y, must equal the negative reciprocal of y y the slope of the tangent line at y., So see if you can solve the system and y.} 49. Find a differentiable function f, which isn t identically zero, that satisfies the integral equation tf t dt f. {Hint: The Fundamental Theorem of Calculus} t F t t f u dudt, then find 5. If f is a continuous function with f and 4 3 the value of F. {Hint: The Fundamental Theorem of Calculus twice}
5. Suppose that f is a continuous function with f d, f d. a) Find a lower bound on the maimum value of f on the interval,? f d {Hint: If you add the three integrals together, you get implies that 3 4 f d, and since, and f d. This f d, we also get that f d. If M is the maimum value of f on, so finish it from here.} f d M d b) Show that the function 3,, then f 36 7 396 48 satisfies the conditions of the problem, and show that its maimum value satisfies the condition of part a). 5. Suppose that f is a differentiable function on the interval, with f and. f. Show that for all in,, f tdt f {Hint: For t, the Mean Value Theorem gives that c t non-decreasing,, which implies that f t tf c tf t, since Integration leads to value of f t on f t f fc for some t f is non-decreasing. f t dt tf t dt M tdt, where M is the maimum,. Finish it from here.}
53. Suppose that f is continuous with f 6 f for all. If value of 8 I f d. 8 8 8 8 f d, then find the. So {Hint: Let u6 to get f d 6 uf 6 udu 6 f d 8 and I I f d 8 6 f d. Add them together, and solve for I.} 3 54. Calculate the value of 3 {Hint: Check-out the graph, d. 3 y 3 3 y dy y 3 3 d and let y in 3 3 y dy.}
55. For a given number a with a, find a non-negative continuous function f with the following properties function eists. {Hint: Multiply the first equation by f d, f d a, f d a a, the second equation by a third equation to get, or show that no such, and add them to the a a f d. This implies that a f d. What does this imply about the function f?} 56. a) Suppose that f is a continuous function on, with an inverse f. Find the value of f f d. f, and f and f y dy f f d f d f y dy. So d. 8 7 b) Find the value of 7 8
57. Suppose that f is a twice differentiable function on, with f f g f and g. Find an upper bound for f on,. d d f f f f f f {Hint: f f f gf, on, This means that f f f f f f f, finish it from here.}, and since