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Calculus Exercises: Limits, Derivatives, and Tangent Lines, Exams of Mathematics

A series of calculus problems focusing on limits, continuity, and derivatives. It includes exercises on evaluating limits from graphs, identifying points of discontinuity and non-differentiability, and applying derivative rules. The problems cover topics such as finding derivatives of trigonometric functions, exponential functions, and composite functions. Additionally, it explores tangent lines and graphical analysis of functions based on their properties. This resource is designed to enhance understanding and proficiency in calculus concepts through practical application and problem-solving.

Typology: Exams

2024/2025

Available from 05/30/2025

paul-marks
paul-marks 🇺🇸

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1.3K documents

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Problem
1:
Use
the
graph
to
answer
the
questions
below.
(@
Jim
f()
=
O
()
lim
o)
IS0
L
\
f(-3)=
DNE
(@
lim
£G)
=
3
(e)
lim
£
(x)
=0ONE
™\
Of=12
(&)
lim
f(x)
=
2
(h)
f(2)
=
DNE
(i)
At
which
x
values
is
f(x)
discontinous?
|,
1
(j)
At
which
x
values
is
f(x)
not
differentiable?
A
|
Problem
2:
|
-F'()()—.\lm
n->0
ey
=
oy
_.——-—--":'"""‘
a.
Exactly
ONE
of
these
limits
represents
the
derivative
of
a
function.
Circle
that
ONE
limit.
(3x2—5x+7+h)—(3x*—5x+7)
i
'l'i_gg
h
.
(Bx*-5x+7+h)+(Bx*-5x+7)
il.
Llixtl)
h
m(3(x+h)2—5(x+h)+7)—(3x2—Sx+7)
Illi—-o
h
iv.
li
i
B
+h)2—5(x+h)+7)+@x2—5x+7)
m
h—0
h
b.
For
ONLY
the
limit
that
you
circled
in
part
(a),
the
limit
represents
the
derivative
of
what
function?
RX):
INTomy
rT
pf3

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Problem 1: Use the graph to answer the questions below.

(@ Jim f() = O () lim^ o)^ IS0^ L
(© f(-3)= DNE (@ lim £G) = 3 (e) lim £ (x) =0ONE ™
Of= (&) lim f(x) = 2 (h) f(2) = DNE (i) At which x values is f(x) discontinous? |, 1

(j) At which x values is^ f(x)^ not^ differentiable?^ —^ A^ |

Problem 2: | -F'()()—.\lm n-> ey = oy _.——-—--":'"""‘ a.^ Exactly^ ONE^ of^ these^ limits^ represents^ the^ derivative^ of^ a^ function.^ Circle^ that^ ONE^ limit. (3x2—5x+7+h)—(3x*—5x+7) i 'l'i_gg h

. (Bx-5x+7+h)+(Bx-5x+7)

il. Llixtl) h m(3(x+h)2—5(x+h)+7)—(3x2—Sx+7)

Illi—-o h

iv. li i B^ +h)2—5(x+h)+7)+@x2—5x+7)

m

h—0 h

b. For^ ONLY^ the^ limit^ that^ you^ circled^ in^ part^ (a),^ the^ limit^ represents^ the^ derivative^ of^ what^ function? RX): INTomy rT

Problem 3: Find the derivative of each of the following (Do not simplify your answer): a.

f(t) = cos(t3) — 5—5 + 3¢

Cos(43)= 342 & 2t

& t 2

—Sn@d) ¢ 42 +19E 0+ 37 N (%) |

VE e i s +an ()

(5x^ ")^ (Fan))^ =^ Secr(xy^ (x'17%)

L^ T^ T^ R

fan Ax) Problem 4: Evaluate the following limits. Show clearly how you got your answer. a.

\ \

== K

3% 3y