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(2*t*), [0, *π*/2]

absolute minimum( ) absolute maximum( )

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

*f*(*t*) = 3*t* + 3 cot(*t*/2), [*π*/4, 7*π*/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^(*−*x*2/72), [−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(2*x*) [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 *x*1 = 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 *x*1 = 0.6 as the initial approximation.

*x* = ( )

(c) Solve the equation in part (a) using *x*1 = 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*(7*π*/2) = 0

*f*(*x*) = ( )

21. Find *f*.

*f* *”*(*t*) = 3*e^t* + 8 sin(*t*), *f*(0) = 0, *f*(*π*) = 0

*f*(*t*) = ( )

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