Math Formal Writing
ahmedalmahmoud
formal_write-up_3.pdf
MTH 251 Name: Summer 2017
Formal Write-up #3 (out of 45 pts) – Due Monday, August 14
Please do the following problems on a separate sheet of paper. A few notes:
• I will be checking for organization, conceptual understanding, and proper mathematical communica- tion, as well as completion of the problems.
• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing.
• Use correct mathematical notation. Include “=” and “≈” where appropriate.
• You may work with your classmates. However, please submit your own work!
• Please use this sheet as a cover page.
1. (Related Rates) A particle moves clockwise around the circle x2 + y2 = 16.
−4 −3 −2 −1 1 2 3 4 5
x
−4
−3
−2
−1
1
2
3
4
5 y
0
(a) In which of the four quadrants is the derivative dy/dt positive? Explain your answer. [3 points]
(b) Find a relation between dx/dt and dy/dt. Show each step and explain you work. [4 points]
(c) At what rate is the y-coordinate changing when the particle passes the point (2 √
2, 2 √
2) if its x- coordinate is increasing at a rate of 3 ft/sec? Explain your answer. [4 points]
(d) What is dx/dt when the particle is at the left-most and the right-most of the circle? That is, at (4, 0) and (−4, 0). Explain your answer. [4 points]
2. Use logarithmic differentiation to find the derivative of the function given by
f(x) = xsin(x).
Show all your work and justify every step! [5 points]
1
3. (Extreme Values) Answer each of the following questions (using complete sentences) in reference to the function f shown below.
(a) What are the critical numbers of f? [3 points]
(b) What are the local extreme points of f? Classify the points as local minimums or local maximums. [3 points]
(c) Find the global maximum value of f over the interval (−7, 7). Is there guaranteed to be a global minimum on the interval (−7, 7)? Discuss why or why not. [4 points]
4. Each of the quantities (a)-(j) can be represented in the picture below. For each quantity, state whether it is represented by a length, a slope, or an area. Then, using letters on the picture, decide which length, slope (i.e., ratio) or area represents it. [Note: The letters P , Q, R, etc. represent points and both `1 and `2 are tangent lines to the graph of f at T and Q, respectively.] [1 point each]
x
y
y = f(x)
a a + h
`1
`2
T
Q
V
P S
R
U
W
X
(a) h (b) f(a + h) (c) f(a + h) −f(a) (d) f(a + h) −f(a)
h (e) f′(a)
(f) f′(a + h) (g) lim h→0
f(a + h) −f(a) h
(h) lim x→a
f(x) (i) f(a) ∗h (j) f′(a) ∗h
2
