A Collection of Problems in Differential Calculus Problems Given At the Math 151 - Calculus I and Math 150 - Calculus I With Review Final Examinations Department of Mathematics,

A Collection of Problems in Differential Calculus
Problems Given At the Math 151 - Calculus I and
Math 150 - Calculus I With Review Final
Examinations Department of Mathematics,
Simon Fraser University
Veselin Jungic · Petra Menz · Randall Pyke
Department Of Mathematics Simon Fraser
University
c Draft date December 6, 2011
Solution by shahbaz ahmed

March 2025


Definitions


Let f be a function defined on some open interval that con-

tains a, except possibly at a itself. Then limx−→a f (x) = L,

=⇒




∀ϵ > 0 ∃ δ > 0 such that | f (x) − L| < ϵ

whenever |x − a| < δ


1

,Prove that

Area of sector of a circle = 21 r 2 x

Where

r=radius of the circle

x=Angle in radian suspended by the corresponding arc

at the center of the circle




2

,Proof

Reference to the figure



3

, OA=OB=r= radius of the circle.

∠ AOB = x radians

Area of circle for 2π radians=πr 2
2
Area of the sector of circle for 2π
2π
radians= πr
2π
2
Area of the sector of the circle for 1 radians= r2
2
Area of the sector of the circle for x radians= r2 x

Prove that

limx−→0 sinx x = limx−→0 sinx x = 1

Proof

Reference to the figure




4