256 STEP 4. Review the Knowledge You Need to Score High

x

y

10

6

x^2 – 36

Figure 11.9-4

Step 2: Find the minimum value ofP.

Enter

yt=x+ 10 ∗x/

(√

(x∧ 2 − 36 )

)

.

Use the [Minimum] function of

the calculator and obtain the

minimum point (9.306, 22.388).

Step 3: Verify with the First Derivative

Test.

Entery 2 =(y1(x),x) and

observe. (See Figure 11.9-5.)

9.306

rel. min.

decr. incr.

• 
  
  
  
  • 
    
  
  
  
  

y 1 = f

y 2 = f′

Figure 11.9-5

Step 4: Check endpoints.

The domain ofxis (6,∞).

Sincex= 9 .306 is the only

relative extremum, it is the

absolute minimum.

Thus, the maximum length of the

pipe is 22.388 feet.

31. 
    

    limx→− 1
    1 +cosπx
    x^2 − 1
    =limx→− 1
    −πsinπx
    2 x

    
    

0

− 2

= 0

32.

dx

dt

=− 4 t

dy

dt

= 2 t. The speed of the object

is

√

(− 4 t)^2 +(2t)^2 =

√

16 t^2 + 4 t^2 =

√

20 t^2 = 2 t

√

5. Evaluated

att=

1

2

, the speed is 2

(

1

2

)√

5 =

√

5.

33. Integrate

∫

2 x

√

x+ 3 dxby parts with

u= 2 x,du= 2 dx,dv=

√

x+ 4 dx, and

v=

2

3

(x+4)^3 /^2. Then

∫

2 x

√

x+ 3 dx=

4

3

x(x+3)^3 /^2 −

4

3

∫

(x+3)^3 /^2 dx=

4

3

x(x+3)^3 /^2 −

4

3

(

2

5

(x+3)^5 /^2

)

. Simplifying this

expression, we get

∫

2 x

√

x+ 3 dx

=

4

5

(x+3)^3 /^2 (x−2).