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There is a function such that f ( x ) < 0 , f ( x ) > 0 , and f ( x ) < 0 . (A graphical “proof” is acceptable for this answer.)

True

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There is a function such that there is both an inflection point and a critical point for some value x = a .

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Given the graph of f , determine where f is increasing or decreasing.

The function increases to cross the x-axis at −2, reaches a maximum and then decreases through the origin, reaches a minimum and then increases to a maximum at 2, decreases to a minimum and then increases to pass through the x-axis at 4 and continues increasing.

Increasing: ( −2 , 0 ) ( 4 , ) , decreasing: ( , −2 ) ( 0 , 4 )

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The graph of f is given below. Draw f .

The function decreases rapidly and reaches a local minimum at −2, then it increases to reach a local maximum at 0, at which point it decreases slowly at first, then stops decreasing near 1, then continues decreasing to reach a minimum at 3, and then increases rapidly.
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Find the linear approximation L ( x ) to y = x 2 + tan ( π x ) near x = 1 4 .

L ( x ) = 17 16 + 1 2 ( 1 + 4 π ) ( x 1 4 )

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Find the differential of y = x 2 5 x 6 and evaluate for x = 2 with d x = 0.1 .

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Find the critical points and the local and absolute extrema of the following functions on the given interval.

f ( x ) = x + sin 2 ( x ) over [ 0 , π ]

Critical point: x = 3 π 4 , absolute minimum: x = 0 , absolute maximum: x = π

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f ( x ) = 3 x 4 4 x 3 12 x 2 + 6 over [ −3 , 3 ]

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Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down.

x ( t ) = 3 t 4 8 t 3 18 t 2

Increasing: ( −1 , 0 ) ( 3 , ) , decreasing: ( , −1 ) ( 0 , 3 ) , concave up: ( , 1 3 ( 2 13 ) ) ( 1 3 ( 2 + 13 ) , ) , concave down: ( 1 3 ( 2 13 ) , 1 3 ( 2 + 13 ) )

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g ( x ) = x x

Increasing: ( 1 4 , ) , decreasing: ( 0 , 1 4 ) , concave up: ( 0 , ) , concave down: nowhere

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Evaluate the following limits.

lim x 3 x x 2 + 1 x 4 1

3

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lim x cos ( 1 x )

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lim x 1 x 1 sin ( π x )

1 π

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lim x ( 3 x ) 1 / x

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Use Newton’s method to find the first two iterations, given the starting point.

y = x 3 + 1 , x 0 = 0.5

x 1 = −1 , x 2 = −1

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Find the antiderivatives F ( x ) of the following functions.

g ( x ) = x 1 x 2

F ( x ) = 2 x 3 / 2 3 + 1 x + C

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f ( x ) = 2 x + 6 cos x , F ( π ) = π 2 + 2

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Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes.

y = 1 x ( x + 1 ) 2


This graph has vertical asymptotes at x = 0 and x = −1. The first part of the function occurs in the third quadrant with a horizontal asymptote at y = 0. The function decreases quickly from near (−5, 0) to near the vertical asymptote (−1, ∞). On the other side of the asymptote, the function is roughly U-shaped and pointed down in the third quadrant between x = −1 and x = 0 with maximum near (−0.4, −6). On the other side of the x = 0 asympotote, the function decreases from its vertical asymptote near (0, ∞) and to approach the horizontal asymptote y = 0.
Inflection points: none; critical points: x = 1 3 ; zeros: none; vertical asymptotes: x = −1 , x = 0 ; horizontal asymptote: y = 0

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A car is being compacted into a rectangular solid. The volume is decreasing at a rate of 2 m 3 /sec. The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of 0.25 m/sec, find the rate at which the height is changing when the length and width are 2 m and the height is 1.5 m.

The height is decreasing at a rate of 0.125 m/sec

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A rocket is launched into space; its kinetic energy is given by K ( t ) = ( 1 2 ) m ( t ) v ( t ) 2 , where K is the kinetic energy in joules, m is the mass of the rocket in kilograms, and v is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of 15 m/sec 2 and the mass is decreasing at a rate of 10 kg/sec because the fuel is being burned. At what rate is the rocket’s kinetic energy changing when the mass is 2000 kg and the velocity is 5000 m/sec? Give your answer in mega-Joules (MJ), which is equivalent to 10 6 J.

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The famous Regiomontanus’ problem for angle maximization was proposed during the 15 th century. A painting hangs on a wall with the bottom of the painting a distance a feet above eye level, and the top b feet above eye level. What distance x (in feet) from the wall should the viewer stand to maximize the angle subtended by the painting, θ ?

A point is marked eye level, and from this point a right triangle is made with adjacent side length x and opposite side length a, which is the length from the bottom of the picture to the level of the eye. A second right triangle is made from the point marked eye level, with the adjacent side being x and the other side being length b, which is the height of the picture. The angle between the two hypotenuses is marked θ.

x = a b feet

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An airline sells tickets from Tokyo to Detroit for $ 1200 . There are 500 seats available and a typical flight books 350 seats. For every $ 10 decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard?

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Questions & Answers

how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
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MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how en where can u apply it
Migos
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
Practice Key Terms 3

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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