Question 1 ( 15 Points):
Find the following limits:
(a) h
h
h
2
1
21
0
lim −
+
→
(b) )5sin(
)4tan(
lim
0x
x
x→
(c) Let )5sin(
)4tan(
)( x
x
xf = for 2/2/
π
π
<
<− x, 0
≠
x. How would you define )0(f so that
)(xf is continuous?
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Limits, Derivatives, and Integrals: Solving Calculus Problems, Exams of Calculus
Calculus problems covering limits, derivatives, and integrals. Students are asked to find limits of functions, equations of tangents, possible values of trigonometric functions, derivatives of functions, and areas under curves. The document also includes problems on the fundamental theorem of calculus and improper integrals.
Typology: Exams
2012/2013
1 / 8
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Find the following limits:
(a)
h
h
h
2
1 2
1
0
lim
→
(b) sin( 5 )
tan( 4 ) lim (^0) x
x
x →
(c) Let sin( 5 )
tan( 4 ) ( ) x
x
f x = for − π / 2 < x < π/ 2 , x ≠ 0. How would you define f ( 0 )so that
f ( x )is continuous?
(a) y = f ( x )is a one-to-one function, and the point (–1, 2) is on its graph. Let ( )
1 f x
− be the
inverse function of f ( x ), and ( ) f ( x ) dx
d f ′^ x = be the derivative of f ( x ). The equation of
the tangent to y = f ( x )at (–1, 2) is y = 2 x + b. Find the following. Justify your answers.
(i) b
(ii) ( 2 )
− 1 f
(iii) f ′(− 1 )
(iv) ( ( 1 ))
1 −
− f f
(v) (^2)
1 ( ) =
− f x x dx
d
(b) If 2
1 sin( x ) =− , then what are all possible values for tan( x )?
(a) Find the (^) ∫
x
x
t dt
dx
d
3 2 using the Fundamental Theorem of Calculus.
(b) Find (^) ∫
x
x
t dt
dx
d
3 2 by first finding (^) ∫
x
x
t dt
3 2 , and then taking the derivative of the result.
(c) Find (^) ∫ +
e
x dx
1
( 2 (ln( ) 1 )) given that the derivative of ln( )
2 x x is 2 (ln( x )+ 1 ).
(a) Evaluate
∫
1
0 , 5
2
2
dx
x
x
(b) Find the area between the curve 2 1
2 y = x x + , 0 ≤ x ≤ 3 , and the x-axis
Determine whether the following sequence is convergent or divergent. If the sequence is
convergent, find its limit.
(a) 1
n
n a
n
n
(b)
n
n a (^) n
ln( + 1 )
For each of the following series, write the first 2 terms and determine whether the series is
convergent or divergent. If the series converges, find its sum.
(a) ∑
∞
=
−
1
( 1 )
n
n
(b) ∑
∞
=
0
1
2
n
n n
n