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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

Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
175000/9*100-100+164294/9*100-100*4
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
66
ram
6
ram
6
Cajab
what is domain in calculus
nelson
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
(-x^5+x^2)/100
Sarada
(-5x^4+2x)/100
Sarada
oh sorry it's (-x^3+1)÷x^2
Misha
-5x^4+2x
Sarada
sorry I didn't understan A with that symbol
Sarada
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
Ga
72x³-72x²+106x-60
okhiria
y= (2x^2+5)(3x+9)^2
lemmor
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
192x^2+50x-1
Daniel
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
Ga
24x²+10x-1
Eyad
eyad Amin that's the correct answer?
Ga
yes
Eyad
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Ga
hahaha 😂❤️❤️❤️ welcome bro ❤️
Eyad
eyad please answer my other question for my assignment
Ga
y= (2x^2+5)(3x+9)^2
lemmor
can i join?
Fernando
yes of course
Jug
can anyone teach me integral calculus?
Jug
it's just the opposite of differential calculus
yhin
of coursr
okhiria
but i think, it's more complicated than calculus 1
Jug
Hello can someone help me with calculus one...
Jainaba
find the derivative of y= (2x+3)raise to 2 sorry I didn't know how to put the raise correctly
Ga Reply
8x+12
Dhruv
8x+3
okhiria
d the derivative of y= e raised to power x
okhiria
rates of change and tangents to curves
Kyaw Reply
how can find differential Calculus
Kyaw
derivative of ^5√1+x
Rohit Reply
can you help with this f(×)=square roots 3-4
oscar
using first principle, find the derivative of y=cosx^3
RUBY Reply
Approximate root 4 without a calculator
Tinkeu Reply
Approximate root 4.02 without using a calculator
Tinkeu
2.03
temigbe
unit of energy
Safiyo
< or = or >4.00000001
Fufa
2.01
Cajab
+-2
PRATICK
find the integral of tan
Gagan Reply
sec
PRATICK
sec
PRATICK
tan
dipesh
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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