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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

find the domain and range of f(x)= 4x-7/x²-6x+8
Nick Reply
find the range of f(x)=(x+1)(x+4)
Jane Reply
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
Harrieta Reply
d/dx{1/y - lny + X^3.Y^5}
mogomotsi Reply
How to identify domain and range
Umar Reply
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
Adri
sorry
Dr
hi adri ana
Dr
:(
Shun
was up
Dr
hello
Adarsh
is it chatting app?.. I do not see any calculus here. lol
Adarsh
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
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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