Question 1: ( 10 Points)
Let f be a function which is thrice differentiable in the interval I. ( '''''' ,, fff exist) . Prove
that ( ))(,( cfc is an inflection point for some
∈
c I if 0)(
'' =cf and 0)(
''' >cf .
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Key points of this exam are: Thrice Differentiable, Function, Interval, Inflection Point, Some, Circle, Radius, Center, Intersected, Implicit Differentiation
Typology: Exams
2012/2013
1 / 6
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Let f be a function which is thrice differentiable in the interval I. ( f '^ , f '', f '''exist). Prove that ( ( c , f ( c ))is an inflection point for some c ∈I if f ''^ ( c )= 0 and f ''^ '( c )> 0.
(a) Show that the circle with radius 1 and center (2,3) will be intersected by the line y = mx + 1 at two points if 0 < m < 0. 75. (b) Using implicit differentiation, show that the line y = mx + 1 intersects the circle at (2.4, 2.8) for m = 0. 75.
Find the following limits:
(a) h
h h h
−⎡^ +
→
(^3) tan( ) (^3) tan( ) lim 0
(b) csc( 3 )
sin( 7 ) tan ( 2 )
lim sec( ) (^0 2) x
x x
x x
→
(c) 2
lim^242 (^2) −
→ (^) x
x x x
Find the derivative f 'at ( x 0 (^) , y 0 )if
(a) f ( x )= cos^2 1 + x^2 , x 0 = 0
(b) tan( )
( ) sin( ) x
f x = x x , / 4
x 0 = π
(c) f ( x )= sin( x −sin( x )), x 0 = 0