# 4.10 Antiderivatives  (Page 7/10)

 Page 7 / 10

There is a function such that $f\left(x\right)<0,{f}^{\prime }\left(x\right)>0,$ and $f\text{″}\left(x\right)<0.$ (A graphical “proof” is acceptable for this answer.)

True

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

Given the graph of ${f}^{\prime },$ determine where $f$ is increasing or decreasing.

Increasing: $\left(-2,0\right)\cup \left(4,\infty \right),$ decreasing: $\left(\text{−}\infty ,-2\right)\cup \left(0,4\right)$

The graph of $f$ is given below. Draw ${f}^{\prime }.$

Find the linear approximation $L\left(x\right)$ to $y={x}^{2}+\text{tan}\left(\pi x\right)$ near $x=\frac{1}{4}.$

$L\left(x\right)=\frac{17}{16}+\frac{1}{2}\left(1+4\pi \right)\left(x-\frac{1}{4}\right)$

Find the differential of $y={x}^{2}-5x-6$ and evaluate for $x=2$ with $dx=0.1.$

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

$f\left(x\right)=x+{\text{sin}}^{2}\left(x\right)$ over $\left[0,\pi \right]$

Critical point: $x=\frac{3\pi }{4},$ absolute minimum: $x=0,$ absolute maximum: $x=\pi$

$f\left(x\right)=3{x}^{4}-4{x}^{3}-12{x}^{2}+6$ over $\left[-3,3\right]$

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

$x\left(t\right)=3{t}^{4}-8{t}^{3}-18{t}^{2}$

Increasing: $\left(-1,0\right)\cup \left(3,\infty \right),$ decreasing: $\left(\text{−}\infty ,-1\right)\cup \left(0,3\right),$ concave up: $\left(\text{−}\infty ,\frac{1}{3}\left(2-\sqrt{13}\right)\right)\cup \left(\frac{1}{3}\left(2+\sqrt{13}\right),\infty \right),$ concave down: $\left(\frac{1}{3}\left(2-\sqrt{13}\right),\frac{1}{3}\left(2+\sqrt{13}\right)\right)$

$y=x+\text{sin}\left(\pi x\right)$

$g\left(x\right)=x-\sqrt{x}$

Increasing: $\left(\frac{1}{4},\infty \right),$ decreasing: $\left(0,\frac{1}{4}\right),$ concave up: $\left(0,\infty \right),$ concave down: nowhere

$f\left(\theta \right)=\text{sin}\left(3\theta \right)$

Evaluate the following limits.

$\underset{x\to \infty }{\text{lim}}\frac{3x\sqrt{{x}^{2}+1}}{\sqrt{{x}^{4}-1}}$

$3$

$\underset{x\to \infty }{\text{lim}}\text{cos}\left(\frac{1}{x}\right)$

$\underset{x\to 1}{\text{lim}}\frac{x-1}{\text{sin}\left(\pi x\right)}$

$-\frac{1}{\pi }$

$\underset{x\to \infty }{\text{lim}}{\left(3x\right)}^{1\text{/}x}$

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

$y={x}^{3}+1,{x}_{0}=0.5$

${x}_{1}=-1,{x}_{2}=-1$

$\frac{1}{x+1}=\frac{1}{2},{x}_{0}=0$

Find the antiderivatives $F\left(x\right)$ of the following functions.

$g\left(x\right)=\sqrt{x}-\frac{1}{{x}^{2}}$

$F\left(x\right)=\frac{2{x}^{3\text{/}2}}{3}+\frac{1}{x}+C$

$f\left(x\right)=2x+6\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x,F\left(\pi \right)={\pi }^{2}+2$

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

$y=\frac{1}{x{\left(x+1\right)}^{2}}$

Inflection points: none; critical points: $x=-\frac{1}{3};$ zeros: none; vertical asymptotes: $x=-1,$ $x=0;$ horizontal asymptote: $y=0$

$y=x-\sqrt{4-{x}^{2}}$

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

A rocket is launched into space; its kinetic energy is given by $K\left(t\right)=\left(\frac{1}{2}\right)m\left(t\right)v{\left(t\right)}^{2},$ 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.

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, $\theta ?$

$x=\sqrt{ab}$ feet

An airline sells tickets from Tokyo to Detroit for $\text{}1200.$ There are $500$ seats available and a typical flight books $350$ seats. For every $\text{}10$ 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?

how to use fundamental theorem to solve exponential
find the bounded area of the parabola y^2=4x and y=16x
what is absolute value means?
Chicken nuggets
Hugh
🐔
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
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
how does this work
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
log tan (x/4+x/2)
Rohan
Rohan
y=(x^2 + 3x).(eipix)
Claudia
Ismael
A Function F(X)=Sinx+cosx is odd or even?
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
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
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
Ok
Endalkachew
how can I solve differentiation?
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?
review of funtion role?