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Math 200 – Calculus I
FINAL EXAM
Fall 2023 The following questions make up the Final Exam. The test is split into two parts. The first part is the multiple-choice part, and the second part is the open answer section. This test is worth 255 points. You have until Wednesday (12/ 13 ) at 11:59 PM to finish the test and email it back to me. Please print off a copy of the test and write all answer on the test. Place all your multiple choice answers on the answer sheet provided on page 12 Please answer all questions on this test. If you need more room to answer the questions use a separate sheet of paper and send it with the rest of the test. You need to send me a copy of the completed test to me via email. Please send me the test in PDF form You only need to send me pages 12 – 24 when you turn in your test. This would include the multiple choice answer sheet (Page 12 ) and answer to open answer section in Part II (pages 13 - 24 ) All work must be your own and you cannot get any help form anyone else except for me. You may NOT talk to other members of the class about the test. If you violate these rules you will get a zero in the class. PLEASE PUT MTH 200 Section W0 1 – FINAL EXAM IN THE TITLE OF YOUR EMAIL WHEN YOU SEND YOUR TEST TO ME
Part I – Multiple Choice.
Answer each question with the best possible choice. Question #1 – Evaluate the following integral 2 1 x dx x
^
a) 3 ln 3 x x b) (^) 2 1 x c x c) 3 3 x c d) 3 ln 3 x x c e) (^) 2 x ln x c Question #2 – If f(x) = 3 x 2 2 x 5 , then find the slope of the tangent line at x = 2. a) 11 b) 14 c) 2 d) 0 e) 2 Question # If f(x) = 2 ln( ) 1 x x , then f '( )x would be a) x 1 x b) (^) 2 1 1 x x 1 c) 1 2 x d) 2 1 x x 1 e) 2 1 x x Question #4 – Find 2 0 lim(2 5) x x a) 0 b) 9 c) 4x d) 5 e) 2 0
Question #10 – Which of the following is the Limit Definition of a Derivative? a)^2 2 1 y y x x b)^2 (^0 2 ) ( ) ( ) h f x f x Lim x x c) ( ) '( ) x a (^) ( ) x a '( ) f x f x Lim Lim (^) g x g x d) 0 ( ) ( ) h f x h f x Lim h e) 0 '( ) '( ) h f x h f x Lim h Question #11 – If f(x) = sin ^1 (2 )x then f '( )x will be…. a) 2 1 1 4 x b) (^) 2 1 1 4 x c) 2 2 1 4 x x d) 2 2 1 4 x e) 2 2 1 4 x
USE THE FOLLOWING GRAPH TO ANSWER QUESTIONS 12 – 16. Question #12 – What is 2 lim ( ) x f x ? a) 2 b)3 c) DNE d) 3 and 2 e) Question #13 – What is (2 ) lim ( ) x ^ f^ x ? a) 2 b)3 c) DNE d) 3 and 2 e) Question #14 – What is 1 lim ( ) x f x ? a) 0 b) 2 c) DNE d) 1 e) Question #15 – What is 4 lim ( ) x f x ? a) 1 b) 2 c) DNE d) 4 e) Question #16 – The function is continuous at which x-values? a) x = 1 b) x = 1 and x = 4 c) x = 3 d) x = 2 e) x = 4 O
Question #20 – Find the instantaneous rate of change of the function f(x) at x = 1? 2 3 ( ) 1 x f x x a) – 9/4 b) 4 c) 9 d) 9/4 e) 3/ Question #21 – If f(x) is continuous at x = 2 and 2 lim ( ) 3 x ^ f^ x then which is true? a) 2 lim ( ) 3 x ^ f^ x b) 2 lim ( ) x f x DNE c) f (2) 3 d) 2 lim ( ) 2 x ^ f^ x e) They are all true Question #22 – Evaluate the following limit… 0 sin(2 ) lim x x x a) 2 b) c) 0 d) 1 e) DNE Question #23 – If 2 f ( )x 2 x tan( )x , then find f '( )x. a) 2 4 x sec( x ) b) 4 x sec( )x c) 2 4 x tan( )x 2 x cot( )x d) (^) 2 2 sec ( ) x x e) 4 x tan( )x 2 x 2 sec ( )^2 x Question #24 – If 2 ln( y) 2 x 4 , then dy/dx would be…… a) 4x + 1/y b) 4x – 1/y c) 4x/y d) 2x + 1/y e) 4xy USE THE FOLLOWING GRAPH TO ANSWER QUESTIONS 25 - 27
Question #2 5 – Which points are inflection points? a) x= - 1 , x = 2, and x = 1/2 b) x = - 1 c) x = 1/ d) x = - 2, x =1/2, and x = 3 e) Only x = 3 Question #2 6 – Which points are relative minimums? a) x = 4 b) x = 2 and x = - 1 c) x = - 2 and x = 3 d) x = e) x = - 1 Question #2 7 – At which points is the graph concave up? a) x = 2 b) x = 4 c) x = - 2 and x = - 1 d) x = 1 and x = 2 e) x = 3 Question #28 – The average rate of change is best described as…. a) the slope of the secant line b) the slope of the tangent line c) the derivative d) the same as instantaneous rate of change e) slope of a horizontal line Question #29 – Evaluate the following integral…
Question # If f ( )x cos( )x x^2 , then 2 2 d y dx would be……. a) - cos(x) + 2 b) - sin(x) + 2x c) - sin(x) d) sin(x) e) cos(x) + 2 Question #34 – Which is true about x = - 1 and the function 3 2 f ( )x 3 x 2 x 5? a) the graph is increasing and concave up b) the graph is increasing and concave down c) the graph is decreasing and concave up d) the graph is decreasing and concave down e) the concavity is zero and the graph is increasing.
Question # If (2 1) ( ) x f x e , then f(x) has a tangent line with the slope of 2 at the point where x is ….. a) x = - 1/2 b) x = 0.307 c) x = 1 d) x = 0.693 e) x = 0. Question #36 – Find the derivative of the following function 2 1 ( ) sin( ) x
f x t dt
Which expression below describes the velocity for the object? a) ' 2 2 f ( )x x sin( x ) b) ' 2 f ( )x 2 x sin( x ) c) ' 2 2 f ( )x x cos( x ) d) f '^ ( )x 2 x cos( x^2 ) e) 3 ( ) cos( 2 ) 3 x f x x
Question #37 – Find all x – values for all inflection points of the function f ( )x x^3 9 x a) x 3 b) x 3 c) x 3 and x 0 d) x 0 e) x^ ^3 and x 0 Question #38 – find the following limit…. . 2 2 2 3 5 lim x 3 2 1 x x x x . a) b) c) 2 d) 0 e) 2/ Question #39 – Find the x-value(s) where the function f(x) = 4 x^3 3 x 1 has a horizontal tangent line. a) x = 2 and x = - 2 b) x = 1/2 c) x =1/2 and x = - 1/ d) x = 0 e) there is no point where the tangent line is horizontal Question # 40 – find dy/dx for the following equation…. ( y 1) 3 x 2 a) 1 b) (^) 2 1 3( y 1) c) 1 3 y d) 1 y END OF PART I - MULTIPLE CHOICE
Part II – Open answer section.
Question #1– Find the following for the function f(x) { 2 points each) 3 2 ( ) 2 12 4 3 x f x x x Relative Min: _____________ Rel Max: __________________ Intervals Increasing: ________________ Intervals Decreasing: _________________ Intervals where concave up: _________________ Intervals where concave down: _______________ X-Values of the Inflection points; ________________ f x^ X2^4
x (^2) 016,0 (^) 2, (^6 0 0 2) V1 (^) 2,
x2 4 12 0 5
(^342515) a treaterghan^ in X 6 2 0 g go^ 012 410 12 12 Dec (^6 ) f x^2 5 7
more an
Inc 5 6 216 4 8
2 4 0 (^3 ) FIT f 3 253 4 10
0th
concavedown
f 0 210
4 4 than
concave damy
f 7 217 4 10
arggter
14 concave
up
Question # ( 3 points each) – Use the function f ( )x 3 x 3 3 x 1 to answer the following questions… a) What is the average rate of change between x = 0 and x = 1? b) What is the instantaneous rate of change at x = 1? c) At what value of x does the function have a horizontal tangent line? 0
f x^9
3
f 1 9 1
(^3 9 ) 6 9 2 (^3 ) (^3 3 2 1 ) 3 2 (^1 0 ) 3 I 5 2 5 E
Question #5 ( 8 points)– Evaluate the following integral….. 2 2 1
^4 x^ x^ ^2 dx
Question #6 ( 7 points) – Find the equation of the tangent line of the function 3 2 f ( )x 3 x 2 x 1 at the point x = 3.
u
(^2 2) du (^) 2x dx dx (^4 )
2 dx^4
Na du 2 u aa
2 ᵗʰ jul 2
211 EE^ 52 2 (^4 ) 524 (^2) 4Th
2 1kg6_ (^4053) (^856 ) F x^9 (^2) 4x
943,32 443 93 y f^3 93 x^3 y 3635 213792
(^93 ) y (^81 18 1 )
y
y 93 181
Question #7 ( 8 points) Evaluate the following integral…. 2 1 x dx
x
a I^ 4 I^ dx^ du S dx^ 5 du (^512 2) du (^) 2n 2in ult (^) C (^21 1) 21N tilt (^) C
2 22N^ IHC
OR
2x (^) 22N tilt (^) C
Question #9 ( 8 points) Find f '( )x for the function 3 2 ( ) sin 2 x f x x u 3 2 g x sin (^) 2x a x (^) Gx
g
x (^2) cos (^) 2x f x^ (^6) sin 2 1 3 2 2 cos^2 sin (^2 ) f x^
Exsin 2 1 6 2 cos
2x sin (^) 2x
Question #10 – Find the following limits… { 3 points each} A) 2 lim 3 x x x e B) 0 2 tan(3 ) lim x 3 x x C) 2 1 2
lim
x 1
x x
x
4 S^ Tex 4 S^ Zex (^0) 4
6sec l
2 6sec 3É^ 6 12 3 (^63 ) 24 (^12 3 ) 4 A DAD
