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Calculus Problems: Limits, Derivatives, and Applications, Exams of Mathematics

A series of calculus problems focusing on limits, derivatives, and applications of calculus. It includes problems on evaluating limits using graphs, computing derivatives of various functions, applying the mean value theorem, implicit differentiation, and optimization problems. The document also covers linear approximation and sketching graphs based on given conditions. It is designed to test and enhance understanding of fundamental calculus concepts and problem-solving skills. This resource is suitable for students studying calculus at the high school or university level, providing a comprehensive review of key topics and techniques. The problems are presented in a clear and structured manner, making it easy for students to follow and practice. The document serves as a valuable tool for exam preparation and reinforcing calculus knowledge.

Typology: Exams

2024/2025

Available from 05/30/2025

paul-marks
paul-marks 🇺🇸

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(32)

1.3K documents

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Problem
1:
(5
points
each)
a.
Use
the
graphs
to
evaluate
the
following
limits.
Te
i
DNE
Jim
)
DNE
nfe
Imhx)
\Hpndekwned
f(x)
\
i
b.
Compute
the
limit,
show
and
justify
your
steps.
2
|z|
r+2
lim
.
W
(Z-¥x
X—>
-7
WL%\
L
R
o
e
pf3
pf4
pf5

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Problem 1: (5 points each) a. Use the graphs to evaluate the following limits.

Te i DNE

Jim ) DNE

nfe

Imhx) \Hpndekwned

f(x) \

i

b. Compute the limit, show and justify your steps.

2 — |z|

r+ lim (^).

W (Z-¥x

X—> -7^ WL%^ L^ R^ o

e

problem 2: (4 points each) For each function £ (x), compute the derivative f'(x). No need to simplify your answers. //,' f(x) = sinh(ln(x)) ] = S —l—— TSR (IRGY Y ¢ gos (^2) (\vxux
f(x) =47 - 3x%)? Tzl (=

  • (a7l ity 4 4x

f(x)_2x2+x+

COERUI s (YU y +1D)

(2{’"rxfl\

f(x) = cos(x?) e?* PRSI0 0+ (^25) (2255 F Los G

  • (029D f(x) = 7 arctan(2x) ] &(\Q: (^7) /;'—’L/»L

problem 5: (10 points) ) T The perimeter i of

a square is increasing at a rate of 2 cm/sec. At what rate is the area

changed ged w when the side i lengths are 2 cm? Provide units in your —————^ answer. Y=Yk p= d

,a_g = ZU’W/%&Q.

Problem^ 6:^ (10^ points)^ For^ the^ function^ f(x)^ =^ xIn(x)^ ... a. Find^ the^ critical^ numbers.^ e : LN^ O^ =D b. Determine^ the^ intervals^ on^ which^ f(x)^ is^ increasing and^ decreasing.^ o U^ e =^ = e^ © |

+ t

xC ————+ (^) > | 0 a \ww.m TeroS (^). (^) () OO NG+, ) (0N

Problem^ 7:^ (10^ points)^ Answ a. Use^ the^ closed^ interval^ method^ to^ fin interval^ [—%,n]. b. A box^ has^ asquare^ base,^ an^ open^ top,^ and a volu er^ ONLY^ ONE^ of^ the following^ questi d the^ maximum ons: and^ minimum^ values of f(x)^ =^ 1+^ cos^ (x) onthe i i^ box that^ requires me^ of^32 m’.^ Find the^ dimensions of^2 q the^ least^ amount^ of^ material. 290 ?)37. ( =

  • i Problem 8: (10 points)^ Use^ either^ linear^ approximation^ OR^ differentials^ to^ estimate^ the^ quantity^ v101. 7 = 10 Y-UE^ LYY W=^ 10=^ M^ -^1000 WO^ ~\Y^ =^ WM^ = SR^ U V