1st Sample Test(1), 1st Semester, 2024
Course title : Calculus1
Department
School year
Lecturer :
ID
이 영
수
검인
Name
DIRECTIONS: Read the directions carefully before beginning.
□
3. Find the limit or show that it does not exist.
Good Luck!
• This exam is a closed book, closed notes. Nothing should be
on the table except a pencil, an eraser, and this exam.
(1)
lim tan
→∞
• NO CALCULATORS or other PAPERS allowed.
• Show all work, clearly and in order.
• Circle your final answers. You will lose points if you do not
circle your answers.
sin
→∞
(2)
lim
(3)
lim tan ln
-------------------------------------------------------------------------1. The figure shows a fixed circle with equation
and a shrinking circle with radius r
and center the origin. P is the point , Q is the upper
point of intersection of the two circles, and R is the point of
→
(sol)
intersection of the line PQ and the x-axis. What happens to R
as shrinks, that is, as → ?
(Ans) (1)
(2)
0
(3)
-----------------------------------------------------------------------4. Determine whether ′ exists.
(1)
sin if ≠
if
(2)
(Ans) The limiting position of R is the point (4, 0).
------------------------------------------------------------------------2. Is there a number that is exactly 1 more than its cube?
sin if ≠
if
(sol)
(Ans) (1) This limit does not exist.
(2) ′
5. The figure shows a point P on the parabola
and the
point Q where the perpendicular bisector of OP intersects the
y-axis. As P approaches the origin along the parabola, what
7. Find the value of c such that the line is tangent
to the curve
.
happens to Q? Does it have a limiting position? If so, find it.
(Ans)
------------------------------------------------------------------------
(Ans) c=6
6. Suppose f is a function that satisfies the equation
---------------------------------------------------------------
for all real numbers x and y. Suppose also that
8. How many tangent lines to the curve pass
through the point (1, 2)? At which points do these tangent lines
touch the curve?
lim
→
(a) Find .
(b) Find ′ .
(c) Find ′ .
(Ans)
(Ans)
(1)
0
(2) 1
(3)
0
