MATH HW
ahmedalmahmoud
formal_write-up_2_2.pdf
MTH 251 Name: Summer 2017
Formal Write-up #2 (out of 30 pts) – Due Tuesday, August 1
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. Given the graph of the function f below, investigate the graph of the derivative f′ (as a function).
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
x
y
f(x)
(a) Where does f have horizontal tangent lines, if any? (By “where” we mean at what points on the domain). [2 points]
(b) On what intervals (of the domain) is f increasing? Decreasing? Use interval notation! [3 points]
(c) Estimate the slopes of several tangent lines in each of these intervals (you do not have to be exact, rough approximations are perfectly ok!). [2 points]
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(d) Using the information you gathered from parts (a)-(c), sketch a graph of f′. You may sketch f′
on this sheet if you wish! [5 points]
2. Let f(x) = xex.
(a) Find f′(x). [2 points]
(b) Find f′′(x). [2 points]
(c) Find f′′′(x). [2 points]
(d) Find f(1000)(x) (i.e., the 1000th derivative of f). [2 points]
(e) Based on parts (a)-(d), find f(n)(x) (i.e., the nth derivative of f). [2 points]
3. A graph of the curve given by the equation y3 − 4y = x2 − 1 is shown in the figure below.
−6 −4 −2 2 4 6
−6
−4
−2
2
4
6
x
y
(a) Why is this curve not a function? Give a specific counterexample. [1 point]
(b) Use implicit differentiation to derive an equation for dy
dx . (Be sure to say where you are using the
chain rule!) [4 points]
(c) Verify the following values by drawing lines onto the figure with the indicated slopes at the indicated points. You may sketch these on this sheet if you wish! [3 points]
dy
dx
∣∣∣∣ (1,2)
= 1
4
dy
dx
∣∣∣∣ (1,0)
= −1 2
dy
dx
∣∣∣∣ (1,−2)
= 1
4
EC Problem (Up to 2 points): Discuss how the differentiability and the continuity of a function re- late. Does one imply the other? That is, does the differentiability imply the continuity? What about the converse? Which condition is stronger/weaker? For the weaker implication, provide a counterexample to show that differentiability (or continuity) does not imply the continuity (or differentiability).
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