Math 180 Exam 1 can you please solve the questions in the pdf please write down your ansewrs on the paper and then sed it to me as a pdf like scan your paper and send it in one pdf use pencil ThankyouMath 180
Exam 1
Name ________________________
Fall 2020
Instructor: Dan Greenheck
Must show your work to receive full credit and (indicate your final answers by enclosing them in a rectangle)
Simplify all your answers whenever possible and only use pencil.
1) (1 pt. each) Use the graph of the function f in the figure to find the following values, if they exist (use
or − when appropriate).
a. lim + f ( x )
b.
c. lim f ( x )
d.
e. lim− f ( x )
f. lim f ( x )
x→ − 1
x→ − 1
lim f ( x )
x→ − 1 −
f ( −1)
x→ − 5
x→ 4
g. lim f ( x )
x → −
2) (3 pts) Using the same graph of f (above, in problem #1) determine the x-values at which f is discontinuous.
3) (8 points)
Sketch the graph of a function f that satisfies all the given conditions.
f (0) = 2
lim f (x) = −
x → 2+
f (1.5) = 0
lim f (x) = −
x→ 2 −
lim f (x) =
x→ − 2 −
lim f (x) = 2
x→
lim f (x) =
x→ − 2 +
lim f (x) = 0
x→ −
4) (5 points each) Find the following limits using algebraic methods. Show all work to receive full credit.
a)
b)
c)
3 − 9− x
x→ 0
2x
lim
lim
x→ −
3x 2 − 4 + 1
5 x + 10
2 x 2 − 18
lim
x→ 3 + 6 − 2 x
5)
(5 points) Considering the theorem for the limit of a composition of functions and the criteria that the
outer function must be continuous at the limiting value of the inner function (assume it is), find
the following limit using all appropriate Limit laws to justify your result.
x −1
lim 3sin −1 2
x→ 1
x − 1
6)
(5 pts. each) Explain why the function is discontinuous at the given number a, using the definition of
Continuity.
3x 2 − 27
,
5 x − 15
a)
f ( x) =
b)
cos x
f ( x) = − 1
x + 1
a =3
x0
if
if
if
x=0 ,
x0
a=0
7)
(6 points) Show that the function f ( x) = 2 x − 5 − 3x is continuous at x = 9 using the definition of
Continuity. Be sure to use the Limit laws where needed.
8)
(7 points)
For the limit lim ( x3 − 2 x + 4) = 3 , find the value of that corresponds to = 0.01. using
x →1
your calculator. Draw a picture of what you found.
9a)
(5 points)
For the limit, lim ( 2 x + 3) = 7 , find the value of that corresponds to = 0.01.
x→2
b)
(7 points)
lim f ( x) = L
x →a
f ( x) − L
10)
Now prove that lim ( 2 x + 3) = 7 using a , argument.
x→2
if given any number 0 we can find a number 0 such that
if
0 x−a
(6 points) Given that lim f ( x ) = 8 & lim g ( x ) = − 2 , determine the limit,
x→2
x→2
using the Limit laws, you must show the use of all laws that apply in your work.
lim
x→2
3 f ( x ) − 5g ( x )
− g ( x)
11)
(11 points) Use the definition of the derivative to find the derivative of the function
f ( x) = 2 x 2 − 3 x
12) (5 points each) Differentiate the following functions using the various rules you have learned.
a)
f ( x) =
2
3
3 x
b)
f ( x) = 3 x 2 e x
5
− 23 x
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