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Calculus problems on integration and differentiation, including finding derivatives, evaluating integrals, and determining limits. It covers topics such as logarithmic functions, trigonometric functions, and inverse functions.
Typology: Exams
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−
t
t
x xedx dt
d
ln
2 1
Question 2: (10 Points)
Given that the derivative of ( 2 1 )
2 e x − x −
x is equal to ( 3 )
2 e x −
x , evaluate the
following integral:
1
0
2 e ( x 3 ) dx
x
(a) Write an integral expression for the volume of the solid generated by revolving the
region bounded by y = –x + 2 , and the lines y = 0, x = 0, and x = 2, about the x-axis.
(b) Write an integral expression for the volume of the solid generated by revolving the
region bounded by y = –x + 2 , and the lines y = 0, x = 0, and x = 2, about y = 3.
(c) Find the volume of the solids in parts (a) and (b) by using the Theorem of Pappus.
Find the derivatives of the inverse functions ( )
1 f x
− of the following functions f ( x ):
(a)
x f ( x )= 2
(b) f ( x )=log 3 ( x )
(c) f ( x )= 2 sin( x )
Find the integrals of the following functions:
(a)
π f ( x )= x
(b)
x
(c) f ( x )= 6 tan( 3 x )
QUESTION 1(15 Points)
Find the following limits:
(a) (^) ⎥
⎦
x → (^) x a + x x cos a
cos( )
lim 0
, given that sec a = 2
(b) sin ( 3 )
tan( 2 )
csc( )
csc( / 2 ) lim (^2) (^0) x
x
x
x
x
→
(c)
2 3
lim
2
→ (^) x
x
x
QUESTION 2(15 Points)
The graph of y = f ( x )is shown below. The x-coordinates of the points A, B … are
x (^) A , x B ….The inflection points are I 1 (^) , I 2 , I 3 , I 4. The tangent to the graph at I 1 is
horizontal, and at I (^) 2 is vertical.
a) What can you say about ( )
' f x at A, B, …, L? Explain!
b) What can you say about ( )
'' f x at A,B,…,L? Explain!
c) Which are the critical points?
d) What is ( )
'' ' f x at I 1 (^) , I 2 , I 3?
