Math 2414 Activity 3(36 parts)(Due July 29)
1. Find the local extrema of the function .
2. You are sitting in a classroom next to the wall looking at the blackboard at the front of the room. The blackboard is 12 ft. long and starts 3 ft. from the wall you are sitting next to.
Your viewing angle is given by . How far from the front of the classroom should you sit to maximize your viewing angle?
Evaluate the following integrals:
3. 4. {Hint: }
5. {Hint: factor the 3} 6. {Hint: Let .}
7.{complete the square} 8.{complete the square}
9. {Hint: Let .}
(9 parts)
10.Evaluate by interpreting it as an area and integrating with respect to y instead of x.
{Hint: .}
11. For , .
So by taking square roots, you get that .
And by taking reciprocals, you get .
Integrate this inequality from 0 to 1 and find upper and lower bounds on the value of .
12. a) Show that, using and completing the long division
.
(3 parts)
b) For , . This means that . Evaluate to see how close is to .
13. Find the inflection point of .
14. Two towns A and B are 8 miles apart. A third town C is located 5 miles from both A and B. If the point P, equidistant from A and B is such that the sum of the distances PA, PB, and PC is the least possible, how far is the point P from C?
The sum of the lengths in terms of the angle is given by .
(3 parts)
15. A 27 ft. ladder is placed vertically against an 8 ft. high fence. The lower end of the ladder is then pulled directly away from the fence. If the ladder is kept in contact with the top of the fence, what is the greatest horizontal distance the ladder ever projects beyond the fence?
The horizontal projection as a function of is given by
.
16. Apply the Mean Value Theorem to the function for to show that .
{ for some c between x and 0.}
17. Use differentiation to show that .
18. Find the angle in the following figure:
{Hint: .}
(4 parts)
19. Verify the following identities:
a) b)
c)
20. If , then show that .
21. Find the local extrema of the function .
22. Evaluate the following integrals:
a) b)
{Hint: .} {Hint: .}
c) d)
{Hint: .}
(9 parts)
23. Assuming that f is increasing and that , use the following diagram to find a formula for involving a definite integral of .
The area of the entire rectangle is .
Use the formula to evaluate the following integrals:
a) b)
c)
24. Show that .
25. Show that any function of the form with , can be written in the form for some constants K and c, by finding formulas for K and c in terms of A and B.
{Hint: .} (6 parts)
26. Use the Bisection Method on the interval to estimate the value of the number c so that the area under the curve between and is equal to 1.
Left(sign) / Midpoint(sign)Estimate / Right(sign) / Maximum Error
1(-) / 1.5(-) / 2(+) / .5
1.5(-) / 1.75( ) / 2(+)
Complete the table and give the best estimate along with the maximum error.
(2 parts)