Find the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible.

1. Find the absolute minimum and absolute maximum values of f on the given interval. f(x) = ((x^2) − 1)^3, [−1, 4]

 absolute minimum( ) absolute maximum( )

2. Find the absolute maximum and absolute minimum values of f on the given interval.

f(t) = 2 cos(t) + sin(2t), [0, π/2]

absolute minimum( ) absolute maximum( )

3. Find the absolute minimum and absolute maximum values of f on the given interval.

f(t) = 3t + 3 cot(t/2), [π/4, 7π/4]

absolute minimum( ) absolute maximum( )

4. Find the absolute maximum and absolute minimum values of f on the given interval.

f(t) =t*sqrt (64-t^2) [−1, 8]

absolute minimum( ) absolute maximum( )

5. Find the absolute maximum and absolute minimum values of f on the given interval.

f(x) = xe^(−x2/72), [−5, 12]

absolute minimum( ) absolute maximum( )

6. Find the absolute minimum and absolute maximum values of f on the given interval.

f(x) = x − ln(2x) [1/2 , 2]

absolute minimum( ) absolute maximum( )

7. Find the dimensions of a rectangle with perimeter 84 m whose area is as large as possible. (If both values are the same number, enter it into both blanks.)

 ( )m (smaller value)

( ) m (larger value)

8. Find the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible. (If both values are the same number, enter it into both blanks.)

 ( )m (smaller value)

( ) m (larger value)

 9. A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is

Y =KN/(9+N^2)

where k is a positive constant. What nitrogen level gives the best yield?

N=( )

10. The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function

P = 120i/(i^2 + i +4)

where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?

i= ( ) thousand foot-candles

11. Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.

Finish solving the problem by finding the largest volume that such a box can have.

V=( )ft^3

12. A box with a square base and open top must have a volume of 4,000 cm^3. Find the dimensions of the box that minimize the amount of material used.

 sides of base =( )m

 height =( )m

13. If 1,200 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

( )cm^3

14. (a) Use Newton's method with x1 = 1 to find the root of the equation

x^3 − x = 4

correct to six decimal places.

 x = ( )

 (b) Solve the equation in part (a) using x1 = 0.6 as the initial approximation.

x = ( )

 (c) Solve the equation in part (a) using x1 = 0.57. (You definitely need a programmable calculator for this part.)

x = ( )

15. Use Newton's method to find all roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

3 cos x = x + 1

 x = ( )

16. Use Newton's method to find all roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

(x − 5)^2= ln(x)

x=( )

17. Use Newton's method to find all real roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

8/x=1+x^3

x=( )

18. A particle is moving with the given data. Find the position of the particle.

v(t) = 1.5*sqrt(t) s(4) = 13

s(t)=( )

19. Find f.

f ''(θ) = sin(θ) + cos(θ), f(0) = 2, f '(0) = 3

 f(θ) = ( )

20. Find f.

f ''(x) = 4 + cos(x), f(0) = −1, f(7π/2) = 0

 f(x) = ( )

21. Find f.

f ''(t) = 3e^t + 8 sin(t), f(0) = 0, f(π) = 0

 f(t) = ( )