MATH 1151 Study Guide - Final Guide: Riemann Sum, Sigma-Algebra, Antiderivative
5 Oct 2018
School
Department
Course
Professor
MSLC – Math 1151
Final Exam Review
1
1. Consider the function
()y fx=
as defined by the graph below.
a) Evaluate the following:
i)
lim ( )
xfx
→−∞
ii)
6
lim ( )
xfx
→−
iii)
2
lim ( )
xfx
→−
iv)
3
lim ( )
x
fx
→
v)
6
lim ( )
x
fx
→
vi)
lim ( )
xfx
→∞
b) What is the domain of
()fx
in interval notation?
(Assume the function goes past the visible portion.)
What is the range of
()fx
?
c) Where is
()fx
discontinuous? Where is
()fx
continuous?
d) Where is
()fx
not differentiable?
e) Estimate
'(1)f
and give the equation of the tangent
line there.
f) Estimate
'( 3)f−
and give the equation of the
tangent line there.
2. Find the following limits or state that the limit does not exist. (Do not use your calculator to help with these!)
a)
2
2
2
4
lim 6
z
z
zz
→
−
+−
b)
lim x
xex
−
→∞
c)
2
0
6
lim sin
x
xx
x
→
+
d)
2
2
0
sin (3 )
lim
t
t
tt
→−
e)
0
1 cos(2 )
lim sin cos
x
x
xx
→
−
f)
2
2
lim 3
u
u
uu
→−∞
+
3. Let
2
2
() 21
x
fx x
+
=+
. Use the definition of the derivative to calculate
'(1).
f
What is the equation of the
tangent line there?
4. Differentiate the following using derivative rules.
a)
33
( ) sin sin( )fx x x= −
b)
( )
2
2 5 ln16
4
() 1
x
gx x e
+
= −
c)
( )
3
54
2
() 2
xx
hx x x
++
= −
5. Let
3
2
( 1) cos
() (csc )
xx
zx xx
+
=
. Find
'( )zx
with logarithmic differentiation.
6. Suppose
32
0x y xy−=
. Find the equations of the tangent lines at the point (1,1).
7. Use implicit differentiation to find
'y
if
2
cos( ) 3xy xy y− +=
.
8. Find the point
(, )xy
on the graph
xy =
nearest the point
(4,0)
.
9. Let
[ ]
( )
2
1 on 0,2
() 1 on 2,
ax bx
fx x
++
=−∞
Find a and b so that
()fx
is continuous and differentiable
everywhere.
75
MATH 1151 Full Course Notes
Verified Note
75 documents
Document Summary
1 y f x as defined by the graph below: evaluate the following: x ii, lim ( ) f x lim ( ) f x x lim ( ) f x x. 6 x f x in interval notation? (assume the function goes past the visible portion. ) lim ( ) f x. 2 x: lim ( ) f x. 3 x: what is the domain of. Where is f x continuous: where is f x not differentiable? f line there. "(1: estimate and give the equation of the tangent, estimate and give the equation of the f . "( 3) tangent line there: find the following limits or state that the limit does not exist. (do not use your calculator to help with these!) 0 t: lim x e x x e) lim x. 1 cos(2 ) x sin cos x x lim x. Use the definition of the derivative to calculate.
