Math 2414 Activity 1 (Due by end of class Jan. 26)

Precalculus Problems:

1. One of the two lines that pass through 3,0 and are tangent to the parabola 2y x is the x-

axis. Find the other line.

{Hint: For a line through 3,0 , 3y m x , to be tangent to 2y x , the system of

equations

2

3

y x

y m x

must have exactly one solution. Find the values of m that

make this true.}

2. Find the center of the circle that passes through the points 1,7 , 8,6 , and 7, 1 .

{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 x mx and g x nx are perpendicular

then 1mn .

{Hint: Apply the Pythagorean Theorem to the following triangle.}

4. If 3 12 2

2 1; 1

; 1

x xf x

x x

and 1

31g x x , then find all values of x so that f x g x .

{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 3 26 3 67

2 3

n n n

n

is an

integer.

{Hint: Complete the long division

2

3 2

3

2 3 6 3 67

n

n n n n

.}

1,m

1,n

y mx

y nx

m n

21 m

21 n

6. If f is a real-valued function with the property that f x y f x f y for all real

numbers x and y , and if 1 3f , then find

a) 2f {Hint: 2 1 1f f .}

b) 3f {Hint: 3 1 2f f .}

c) 4f {Hint: 4 2 2f f .}

d) 1,000,000f

e) 12

f {Hint: 1 1 1 12 2 2 2

1f f f f .}

f) 0f {Hint: 1 1 0 1 0f f f f . }

g) 1f {Hint: 0 1 1 1 1f f f f . }

7. If f is a real-valued function with the property that f x y f x f y for all real

numbers x and y , and if 1 3f , then find

a) 2f {Hint: 2 1 1f f .}

b) 3f {Hint: 3 1 2f f .}

c) 4f {Hint: 4 2 2f f .}

d) 1,000f

e) Show that 0f x for all real numbers, x. {Hint: 2

2 2 2x x xf x f f .}

f) 12

f {Hint: 1 1 1 12 2 2 2

1f f f f .}

g) 0f {Hint: 1 1 0 1 0f f f f . }

8. Given that 11 11f and

13

1

f xf x

f x

for all x, find 2015f .

{Hint: First find 14 , 17 , 20 ,f f f ., and look for a pattern.}

Graph of f Graph of g

Graphing, Limits and Continuity Problems:

9. Using the graphs of the functions f and g, determine the following:

a) 3f g b) 1f g c) 2f g d) 3f

g

e) 2f

g

f) 3f g g) 0g f h) 2g g

i) 2f g g j) Solve 1f g x . k) Solve 0f g x .

l) Solve 1f g x . m) xgxfx

0

lim n) 1

limx

f x g x

o) 1

limx

f x g x

p) 1

limx

f x g x

q) 1f g

r) Is f g continuous at 1x ? s) 2

limx

f x g x

t) 2

limx

f x g x

u) 2

limx

f x g x

v)

xg

xf

x 2

lim

w)

xg

xf

x 2

lim x)

xg

xf

x 2lim

y)

xg

xf

x 1lim

z) xgfx 1lim aa) xgf

x 1lim bb) xgf

x 3lim

10. Show that the equation 3 15 1 0x x has exactly three solutions in the interval 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?}

11. Find values of a, b, and c that will make 2

1 , 1

,1 2

3 ,2 3

6 , 3

x x

ax bx c xf x

x x

cx x

continuous.

{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 1,2,3x by solving

the system of equations

2

4 2 6

9 3 6

a b c

a b c

c

.}

12. If lim 2x a

f x g x

, lim 1x a

f x g x

, then find

a) limx a

f x

b) limx a

g x

c) limx a

f x g x

{Hint: 1

2f x f x g x f x g x ,

1

2g x f x g x f x g x . }

13. Let

2

1

11

1 x

f x

.

a) Find 0

limx

f x

b) Find 2

limx

f x

c) Find 1

limx

f x

{Hint: 2

1 2 21 1 ; 0

1 2 2x

x xx

x x

.}

14. Let 2 7 12x x

f xx a

.

a) For what value(s) of a, does limx a

f x

equal a number?

b) For what value(s) of a, does limx a

f x

?

c) For what value(s) of a, does limx a

f x

?

{Hint: 2 7 12 3 4x x x x .}

Derivative Problems:

15. Show that the function 5 2f x x x 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.}

16. Find constants a, b, c, and d so that the graph of 3 2f x ax bx cx d has horizontal

tangents at 0,1 and 1,0 . {Hint: Make the derivative and function values correct.}

17. Here’s the graph of the function f on the interval 4,3 . It has a vertical asymptote at

3x ,

xfx 3lim .

a) What are the critical numbers of f ? b) What is the absolute maximum of f on

4,3 ?

c) What is the absolute minimum of f on 4,3 ? d) Where does f have local maxima?

e) Where does f have local minima? f) Find the maximum of f on 0,2 .

g) Find the minimum of f on 3,1 . h) Find the maximum of f on 2,0 .

i) Find the minimum of f on 4,2 . j) Find the minimum of f on 2,3 .

k) Where is the function increasing? l) Where is the function decreasing?

18. 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 maxima?

d) Where does f have local minima?

e) Which is larger 0f or 1f ?

f) Which is larger 1f or 0f ?

g) Does f have an inflection point at 2x ?

h) Does f have an inflection point at 2x ?

i) Which is larger, 2f or 2f ?

{Hint:

2

2

2 2f f f x dx

.}

19. If 30

1

1

n n

n

f x x x

, then find f x . {Hint: Simplify the sum first.}

20. Find f x if it is known that 22d

f x xdx

.

{Hint: Differentiate 2f x using the Chain Rule, then replace x with 2x .}

f

21. Here is the graph of f for a function f which is continuous on the interval 0,8 .

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 maxima.

f) Find the local minima. g) Find the inflection points.

22. Using the graphs of the functions f and g, evaluate the following:

a) 12

f g b) 12

f g c) 2g f

d) 14

f f e) 32

g g f) 1g f

g) 2f g g h) 52

f g f i) 14

g g g

Graph of f Graph of g

23. Find equations for all the lines through the origin that are tangent to the parabola 2 2 4y x x .

{Hint: At such a point, 2 2 4 0

2 20

x xx

x

.}

24. If 1 2f , then find the derivative of

a) 2f x when 12

x . b) 1f x when 0x . c) 14

f x when 4x .

25. Let 2 cosf x x kx for 0k .

a) There is a number A so that if 0 k A , then f has no inflection points, and if k A ,

then f has infinitely many inflection points. Find A.

{Hint:

2 2 22 2 cos 2

f x

k k kx k

.}

k

k

2

b) Show that f has only finitely many local extrema no matter what the value of k.

{Hint:

2 2 sin 2

f x

x k x k kx x k

, so what can you say about f x if 2kx or

2kx ?}

Rolle’s and Mean Value Theorem Problems:

26. Is there a value of a so that 3 3f x x x a has two distinct zeros in the interval 1,1 ?

{Hint: If there is such a value of a, then Rolle’s Theorem implies that the derivative must

equal zero in the interval 1,1 .}

27. Show that the function 7 5 3 1f x x x x has exactly 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.}

28. Suppose that f exists on ,a b , 0f a f b , but 0f x on ,a b . Show that

0f x on ,a b .

{Hint: Suppose that 0f c for some c in the interval ,a b , and use Rolle’s Theorem as

in the previous hints.}

2y x

k

k

29. Suppose that 0 0f and f x is increasing. Show that the function f x

g xx

is

increasing on 0, .

{Hint:

2

f xf xxf x f x xg x

x x

and the Mean Value Theorem implies that

0

0

f x ff c

x

for some c in 0, x . But this means that

f xf c

x for

some c in 0, x . Take it from here.}

30. Is there a differentiable function f with 1 7f , 3 2f , and 1f x on 1,3 ? If

the answer is yes, then give an example of such a function. If no is the answer, explain why

such a function is impossible.

{Hint: Mean Value Theorem}

Approximating Solutions by Iteration Problems:

31. Attempt to solve the following equations using a cobweb diagram with the given starting

value:

a) 1 6, 2 3 2x x x b) 1 18, 2 3 2x x x

lim nn

x

lim nn

x

c) 1 15, 6x x x d) 211 5

6,x x x

lim nn

x

lim nn

x

e)

2

1

2 10,

4

xx x

f)

2

1

2 14,

4

xx x

lim nn

x

lim nn

x

{Check out the Successive Approximation Worksheet link on the course

webpage!}

Integration and Fundamental Theorem Problems:

32. Let f be the function graphed below on the interval [0,13]. Note: The graph of f consists of

two line segments and two quarter-circles of radius 3.

a) Evaluate 2

0

dxxf b) Evaluate 5

0

dxxf c) Evaluate 8

2

dxxf

d) Evaluate 8

2

dxxf e) Evaluate 13

8

dxxf f)Evaluate 13

0

dxxf

g) If x

dttfxF

0

, then construct the sign chart for the derivative of F on the interval

13,0 .

{Hint: Use the Fundamental Theorem of Calculus: x

a

df t dt f x

dx

.}

h) Find the local extrema and absolute extrema for F on the interval 13,0 .

i) If x

dttFxH

0

, then construct the sign chart for the derivative of H on the interval

13,0 .

33. Is there a differentiable function f so that 2f x and sinf x f x x ?

{Hint: For 0x , 2 2 21 1 12 2 20

0

0

xx

f t f t dt f t f x f and 0 0

sin

x x

f t f t dt t dt ,

so integrate , plug in , and see what happens.}

34. Use the fact that 4 4

1 1 1;1 2

17 1x

x x

to find good upper and lower bounds for

2

4

1

1

1dx

x .

{Hint: If g x f x h x for a x b , then b b b

a a a

g x dx f x dx h x dx .}

35. Find all continuous functions f on 0,1 so that 1

0

x

x

f t dt f t dt for all x in 0,1 .

{Hint: Use the Fundamental Theorem of Calculus: x

a

df t dt f x

dx

.}

36. If for 0x ,

2

525

0

x

tf t dt x , then find the value of 4

5f

.

{Hint: Use the Fundamental Theorem of Calculus: x

a

df t dt f x

dx

.}

37. If f and g are continuous functions, then what is the value of

2

2

f x f x g x g x dx

?

{Hint: For F x f x f x , notice that F x f x f x F x , so F is an

even function. Try the same thing with G x g x g x .}

38. Suppose that f is continuous and 1

0

2f x dx , 2

0

1f x dx , and 4

2

7f x dx .

a) Find the value of 4

0

f x dx . b) Find the value of 0

1

f x dx .

c) Find the value of 2

1

f x dx .

d) Explain why there must be an 0x in 1,2 with 0 0f x .

{Hint: Suppose that 0f x for x in 1,2 , then what can you say about 2

1

f x dx ?}

e) Explain why there must be an 0x in 2, 4 with 0 3.5f x .

{Hint: Suppose that 3.5f x for x in 2,4 , then what can you say about 4

2

f x dx ?}

39. Suppose that f and g are continuously differentiable functions with f x g x for all x.

Show that there must be a number a so that 1f a g a .

{Hint: Suppose the opposite is true: 1f x g x for all x, then for 0x ,

0 0

x x

f t dt g t dt x . This implies that 0 0f x g x x g f . What

happens if 0 0x g f ?}

40. Show that the function

1

2 2

0 0

1 1

1 1

xx

f x dt dtt t

, for 0x , is a constant function.

{Hint: What’s the derivative of this function?}

41. a) If f is continuous and 4

0

10f x dx , then find 2

0

2f x dx .

{Hint: Make a substitution.}

b) If f is continuous and 9

0

4f x dx , then find 3

2

0

xf x dx .

{Hint: Make a substitution.}

c) If f is continuous and 5

2

10f x dx , then find 2

1

3 1f x dx .

{Hint: Make a substitution.}

d) If f is continuous and 1

0

3f x dx , then find 1

0

1f x dx .

{Hint: Make a substitution.}

e) If f is continuous and 1

0

3f x dx , then find

32

1

3 2f x dx .

{Hint: Make a substitution.}

42. The graph of the derivative f x is given. Fill in the table of values for f x given that

0 2f .

{Hint: 0

0

x

f x f t dt f .}

x 0 1 2 3 4 5 6

f x 2

43. If 3 2

b

a

f x dx a b , then find the value of 2 1

a

b

f x dx .

44. If 2

2

0

2 3

x

u

t

f x t e du dt , then find the value of 2f .

{Hint: The Fundamental Theorem of Calculus twice}

45. If f is continuous on 2,4 ,

2

1

2 10f x dx and 2 4f , then find 4f .

{Hint: Let 2u x in the integral.}

46. Suppose that f is a continuous function on 0,1 with 1

0

1

2f x dx . Show that there must

be an 0x in 0,1 with 0 0f x x .

{Hint: Suppose that f x x for all x in 0,1 and arrive at a contradiction. Do the same

with f x x . Or apply the Mean Value Theorem to the function

0

x

F x f t t dt on the interval 0,1 .}

Without assuming continuity, there doesn’t have to be an 0x in 0,1 with 0 0f x x .

Consider 12

12

1;0

0; 1

xf x

x

.

1

0

1

2f x dx

From the graph, you can see that there is no 0x in 0,1 with 0 0f x x .

47. Find the points on the hyperbola 2 2 2x y that are closest to the point 0,1 .

{Hint: The slope from 0,1 to the closest points ,x y must equal the negative reciprocal of

the slope of the tangent line at ,x y . So see if you can solve the system 1

0

y y

x x

and

2 2 2x y .}

48. Find a differentiable function f, which isn’t identically zero, that satisfies the integral

equation 2

0

x

tf t dt f x .

{Hint: The Fundamental Theorem of Calculus}

49. If f is a continuous function with 2 1f and 2

0 2

4 3

x t

F x t t f u du dt

, then find

the value of 2F .

{Hint: The Fundamental Theorem of Calculus twice}

0,1

,x y