Winter 2006 Math 206 FINAL Name:__________________ email:____________________ Show all NECESSARY work NEATLY, CLEARLY, SYSTEMATICALLY for full-point. Any wrong statement and/ understatement may be penalized. Be explicit and box your answers. I’ll consider 200 of 230 points available. Scoring 130 will be a direct pass, a minimum of C. 1. (7) For what values of C is the function f(x) continuous everywhere?

⎪⎩

⎪⎨⎧

>−

≤+=

31

31)(

2 xcx

xcxxf

So, c = _____ 2. (10) Find the absolute maximum and absolute minimum of xxexf 3)( −= on [-1, 1]. So, maximum ________ at x = ________; minimum ________ at x = _________ 3. (15;3,3,5,4) Use linear approximation to estimate 3 26 . You need to first state the function you use, the

point of estimate, and the differential, before plug in numbers.

4. (Total of 35, 5 each) Compute the limit:

a. 24lim

4 −

−→ x

xx

b. xxx

x sin3lim

2

0

+→

c. 32

9lim 2

2

3 −+

−−→ xx

xx

d. x

xx

x eee

2

23

lim −−∞→

e. xxx

3cot7lim 22

0→

f. 32

9lim 2

2

−+

−−∞→ xx

xx

g. x

x x⎟⎠

⎞⎜⎝

⎛ +→

21lim0

. Hint: Let x

xy ⎟

⎠

⎞⎜⎝

⎛ +=21ln

5. (Total of 49; 7 each) Find the derivative y’. You don’t need to simplify:

a. x

xxy 1322 −

+=

b. xxy 3cos2=

c. ( )33sinln xxy +=

d. xy1sin3−

=

e. ( )21 31tan xy −= −

f. xxy

sec1tan+

=

g. xxy sin= . Hint: Use logarithmic differentiation.

6. (Total 22) Consider the following function and its first and second derivative:

4)( 2 +=xxxf

22

2

)4(4)('

+

−−=xxxf

32

2

)4()12(2)("

+

−=

xxxxf

Remember: x2 + 4 > 0

a. (3) Find the asymptote of f(x).

b. (4) Find the critical numbers of f(x). [Note: not critical points, but just critical numbers]

c. (3) Determine the interval where f(x) is increasing.

d. (3) Find the local maxima and local minima of f(x).

e. (6) Find the inflection points of f(x).

f. (3) Determine the interval where f(x) is concave down.

7. (18) Use implicit differentiation to find the equation of the tangent line at (0,0) to the curve )cos()cos( 22 yxxy =

Note: The implicit differentiation is pretty long. But it will clean up once you plug in (0,0)

8. (Total 14) Let ⎪⎩

⎪⎨⎧

=

≠⋅=

00

cot)(

2

x for

0 x forxxxf .

a. (6) Show if f(x) is continuous at x = 0.

b. (8) Show if f(x) is differentiable at x = 0.

9. (Total 8) Consider cos x + sin x = 4x. Show that the equation has one unique solution by a. (4) Arguing that the equation has AT LEAST one solution. (Hint: Use IVT)

b. (4) Arguing that the equation has AT MOST one solution. (Hint: Show one-to-one) 10. (25;3,4,2,5,4,2,2,3) A cylindrical open-top can is to be constructed so that its volume will be 54π in3. If the

cost of material used to make the bottom (again, no “top”) is $2/in2 and the cost of the material for the lateral side is $1/in2, find the radius r and the height h of the least expensive can, then compute the cost of material for that can. Do in the following order: 1>Write r in term of h, 2>Set up the price function in terms of r and h, 3>Rewrite the function in term of r only, 4>.Find the critical numbers, 5>Do sign graph, get the r, 6>Get the h, 7>Get the least price.

11. (Total 13) An upside-down conic water-tank has a circular top of radius 2 m and 9 m high. If the water-

level is denoted h, the radius of the surface of the water is r. hrhrV 2

3),( π=

a. (3) Write h in terms of r.

b. (10) Suppose the tank is leaking at a constant rate of 2 m3/min. Find the rate of decrease of the water-level in the tank when it is 3 m deep.

12. (Total 12) Consider 1254)( 35 +++= xxxxP .

a. (3) Show that P(x) is differentiable everywhere by finding P’(x) and explain why its domain is all real number.

b. (3) Show that P(x) is one-to-one.

c. (3) Find P-1(-10).

d. (3) Find ( ) )10('1 −−P

2 m

9 m r

h