fall 2016: calculus i practice midterm i - math.columbia.edupicard/mid1_prac.pdf · fall 2016:...

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