Math 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.

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1 Math Activity (Due by end of class Jan. 6) 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 equations y make this true.} y m 3 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.}.

2 3. Prove that if the graphs of the linear functions f then mn. m and g n are perpendicular {Hint: Apply the Pythagorean Theorem to the following triangle.} y m m, m m n n, n y n. If f 3 ; ; and g, then find all values of so that f g 3. {Hint: Set the two formulas for f equal to g, and solve carefully.} 5. Use polynomial long division to find the largest integer n, so that integer. {Hint: Complete the long division 3 6n n 3n 67 n 3 is an 3n.} 3 n 3 6n n 3n 67

3 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 {Hint: f 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 {Hint: f 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, find f f 5. {Hint: First find, 7,, f f f., and look for a pattern.}

4 Graphing, Limits and Continuity Problems: 9. Using the graphs of the functions f and g, determine the following: Graph of f Graph of g f g 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. m) lim f g n) lim f g o) lim f g p) f g q) f g lim r) Is f g continuous at? s) lim f g

5 t) lim f g u) f g lim v) lim f g w) lim f g ) f lim y) g lim f g z) lim f g aa) lim f g bb) lim f g 3 3. Show that the equation 5,. {Hint: Find three sign changes in the interval,, and use the Intermediate Value Theorem. How many real zeros can a third degree polynomial have?} 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. Find values of a, b, and c that will make f the system of equations a b c 6.} 9 3c 6. If lim f g, f g a a) lim f a lim, then find a b) lim g a, g f g f g. } {Hint: f f g f g lim f g a c)

6 3. Let f. a) Find lim f b) Find lim f c) Find lim f {Hint: ;.}. Let f 7. a a) For what value(s) of a, does lim f a 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.} 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.} 5. Show that the function 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.}

7 7. Here s the graph of the function f on the interval 3, 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,? c) What is the absolute minimum of f on 3,? 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,. j) Find the minimum of f on 3,. k) Where is the function increasing? l) Where is the function decreasing?

8 8. 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 9. 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

9 . Here is the graph of 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.. 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) g f g) f g g 5 h) f g f i) g g g

10 3. Find equations for all the lines through the origin that are tangent to the parabola y.. If f, then find the derivative of a) f when. b) 5. Let f cosk {Hint: At such a point, f when. c).} f when. 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: f k k

11 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,.} 6. Is there a value of a so that Show that the function f has eactly one real zero. {Hint: Find a sign change, and use the Intermediate Value Theorem to conclude that it has at least one real zero. Suppose it has two or more, and use Rolle s Theorem as in the previous hint.} 8. 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

12 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.} 9. 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

13 c) 5, 6 d) 6, 5 lim n n lim n n e), f), lim n n lim n n {Check out the Successive Approimation Worksheet link on the course webpage!}

14 Integration and Fundamental Theorem Problems: 3. 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.

15 f f? 33. 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, 3. Use the fact that ; 7 d. 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 35. 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 36. If for, tf t dt 5 5, then find the value of f 5. d.} d a {Hint: Use the Fundamental Theorem of Calculus: f tdt f 37. 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

16 38. Suppose that f is continuous and f d, f d, and 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, with f 3.5. d?} {Hint: Suppose that f 3.5 for in,, then what can you say about f d?} 39. 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. Show that the function, for t t f dt dt, is a constant function. {Hint: What s the derivative of this function?}

17 . a) If f is continuous and f d, then find f 9 b) If f is continuous and d. f d, 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.}. The graph of the derivative f is given. Fill in the table of values for f f. given that f.} {Hint: f f tdt f

18 b 3. If f d 3a b, then find the value of a. 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} 5. If f is continuous on,, f d and f, then find 6. 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 the Mean Value Theorem to the function on the interval F f t t dt,.} 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.

19 7. 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.} 8. 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 9. If f is a continuous function with f and 3 the value of F. {Hint: The Fundamental Theorem of Calculus twice}