# Quiz # 3 Math 101-Section 09 Calculus I 6 October ```Quiz # 3
Math 101-Section 09 Calculus I
6 October 2015, Tuesday
Bilkent University
Department of Mathematics
Instructor: Ali Sinan Sertöz
In this quiz you can use only pencils and erasers.
Q-1) We have a function defined as
(
f (x) =
1−cos x
x
0
if x 6= 0,
if x = 0.
(i) Calculate f 0 (0).
(ii) Write an equation of the tangent line to the curve y = f (x) at x = 0.
(iii) Calculate f 0 (π).
(iv) Write an equation of the tangent line to the curve y = f (x) at x = π.
: Grading is 25 points for each question.
1−cos x
−0
f (x) − f (0)
1 − cos x
1
= lim x
= lim
= .
2
x→0
x→0
x→0
x
x
x
2
0
An equation of the tangent line at x = 0 is given as y = f (0)(x − 0) + f (0), so it is written as
f 0 (0) = lim
y = (1/2)x.
On the other hand we have for x 6= 0,
0
1 − cos x
(sin x)(x) − (1 − cos x)(1)
0
f (x) =
=
.
x
x2
Hence we have f 0 (π) = −
2
. And an equation for the tangent line is given by
π2
y=−
where we used f (π) = 2/π.
2
2
(x − π) + ,
2
π
π
```
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