fall 2016: calculus i practice midterm i - math.columbia.edupicard/mid1_prac.pdf · fall 2016:...
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Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
• Answer the questions in the spaces provided on the question sheets.If you run out of room for an answer, continue on the back of the page.
• NO calculators or other electronic devices, books or notesare allowed in this exam.
• Please make sure the solutions you hand in are legible and lucid.You may only use techniques we have developed in class.
• You will have 75 minutes to complete this exam.
Question Points Score
1 10
2 20
3 10
4 10
5 10
6 10
Total: 70
Page 1
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
1. Consider the function
J(x) =
{x2 if x > 0 and x is an integer12
otherwise
(For example, J(1) = 1, J(2) = 4, J(3) = 9, J(4) = 16, but J(0) = 12, J(−7/2) = 1
2,
J(e) = 12, etc.)
(a) 5 points State the domain of
f(x) =1
J(x)− 9.
.
(b) 5 points Sketch the graph of
f(x) = cos(πJ(x)),
on [0, 4].
Page 2
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
2. Find the limit, if it exists. If the limit does not exist, explain why.
(a) 5 points
limx→0
1
xcos(1/x)
(b) 5 points
limx→∞
3x3 + 2x+ 4√9x6 + x2 + 1
Page 3
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
(c) 5 points
limx→0
1
x√x+ 1
− 1
x
(d) 5 points
limx→0+
1
lnx
Page 4
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
3. 10 points Consider the function
J(x) =
{x2 if x > 0 and x is an integer12
otherwise
Find the limit, if it exists. If the limit does not exist, explain why.
limx→∞
(1− J(
√x)
x2
)(2x+ 1
3x− 2
).
Page 5
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
4. Consider
f(x) =
{x cos(1/x) if x > 00 if x ≤ 0
(a) 5 points Is the function f(x) continuous at x = 0?
(b) 5 points Is the function f(x) differentiable at x = 0?
Page 6
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
5. Consider the functionf(x) = x|x|.
(a) 5 points Show that f is differentiable at x = 1 and compute f ′(1) using thedefinition of the derivative.
(b) 5 points Show that f is differentiable at x = 0 and compute f ′(0) using the defi-nition of the derivative.
Page 7
Fall 2016: Calculus IPractice Midterm I Name (Last, First)/UNI:
6. 10 points Show that the following equation admits a solution x.
lnx+ 10x = 0.
Page 8