# Calculus

calculus
To ﬁnd the length of the curve deﬁned by y = 5x^3+9x from the point (-3,-162) to the point (1,14), you’d have to compute the integral [a,b] f(x)dx where a = b = f(x) =

calculus
A cable hangs between two poles of equal height and 40 feet apart. At a point on the ground directly under the cable and x feet from the point on the ground halfway between the poles the height of the cable in feet is h(x) = 10+(0.1)*(x^1.5). The cable weighs 19.2 pounds per ...

Calculus
Consider the differential equation dy/dx = x^4(y - 2). Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 0. Is this y=e^(x^5/5)+4?

Calculus
Write an expression for y = f(x) by solving the differential equation dy/dx = x√y with the initial condition f(3) = 25. I got y = (x^2/4 + 11/4)^2.

Calculus AP Exam review explanation pls
1)What is the area bounded by y = x^2 and y =3x? A)5 B)9/2 ***C)8 D)11.2 E)25 i believe it to be 8 but im not sure. 2)The region R is bounded by the x-axis, x = 2, and y = x^2. Which of these expressions represents the volume of the solid formed by revolving R about the line x...

calculus
dt/dx= ((x^2+a^2)^(1/2))/v1+((b^2+d^2)^(1/2))/v​1 the function dt/dx has a zero at a unique x on (0,d). Use this to justify your discovery that d= (x^2+a^2)^(1/2) + (b^2+d^2)^(1/2)

calculus
find dt/dx t=((a^2+x^2)^1/2)/v1 + ((d^2+b^2)^1/2)/v2

Pre calculus
ramp needs an angle of elevation no greater than 4.8. the business has 8 ft to build the ramp. the ramp must rise 6 inches above the ground. can a ramp be built in this space? if a ramp can be built what is the minimum horizontal distance possible? please show work. i'm ...

Calculus check 1 questiom
Determine the equation the tangent line on curve 4e^x on (0,4). Since derivative of 4e^x =4e^x, then I plugged in 0 for X and the slope I got was 4. Then I wrote the equation and got y=4x +4

Calculus
The position of a particle moving along the x-axis as a function of time,t, is given by x(t)=(1/6)t^3-t^2+3t-1 for t≥0. The particle's velocity becomes three times its initial velocity when t=? I know v(t)=x'(t)=(1/2)t^2-2t+3=9, but I do not understand where does the 9 come ...

Calculus
At which point on the curve y = -2+2e^x is the tangent line parrellel to the line 3x-y=5? Just give the x-coordinate as an exact number

calculus-snell's law
Suppose A light ray starts at the point A = (0,a) in an uniform medium 1 where the speed of light is c1 and then passes through an uniform medium 2 where the speed of light is c2 reaching point B = (d, −b). The line separating the two media is the x-axis; HINTS ONLY 1) what ...

calculus
Find the volume of the solid obtained by rotating the region bounded by the curves y = x^8, y = 1 about the line y = 5.

calculus
Find the volume of the solid obtained by rotating the region bounded by the given curves about the speciﬁed axis. x+y = 3, x = 4−(y−1)^2; about the y-axis.

calculus
Find the volume of the solid obtained by rotating the region bounded by the given curves about the speciﬁed line. x = 1−y^4, x = 0; about x = 1.

calculus
Find the volume of the solid obtained by rotating the region bounded by the given curves about the speciﬁed axis. y = 64x−8x^2, y = 0; about the y-axis.

calculus I
Problem: Consider (1) the parabola y=3-1/10 x^2 and (2) the upper half of the circle centered at (20, 0) with radius of 10. Find the points on the parabola where the tangent line is also tangent to the upper half of the circle. (You can find these points in exact form in ...

calculus
Find the volume of the solid obtained by rotating the region bounded by the given curves about the speciﬁed line. x = 1−y^4, x = 0; about x = 1.

Calculus
The region enclosed by the graph e^(x/2), y=1, and x=ln(3) is revolved around the x-axis. Find the volume of the solid generated. I don't understand if we have to use the washer method or the disk method for this one because when I drew it out on a graph it looked very confusing.

calculus- optimization
A rectangle is inscribed into a semi circle at radius 2. What is the largest area it can have and what are the dimensions Answers Area= 4 max base =2sqrt2 height = sqrt2 Help is always appreciated :)

Calculus
The region enclosed by the graph e^(x/2), y=1, and x=ln(3) is revolved around the x-axis. Find the volume of the solid generated. I don't understand if we have to use the washer method or the disk method for this one because when I drew it out on a graph it looked very confusing.

Calculus
The length l of a rectangle is decreasing at the rate of 3cm/sec, while its width w is increasing at the rate of 3cm/sec. Find the rates of change of the perimeter, and the length of one diagonal at the instant when l=15 and w=6.

Calculus Reposted
A point is moving along the curve xy=12. When the point is at (4,3), the x-coordinate decreases at the rate of 2cm/sec. How fast is the y-coordinate changing at that point? dy/dx=-y/x is my change rate so far, should i substitute the coordinates (x,y) to the equation or just ...

Calculus
Suppose that the price p, in thousand pesos, and the number of sales x(in hundreds) of a certain item can be modeled by the equation 5p+4x+px=100. Suppose also that the price is increasing at the rate of \$200 per year. How fast is the quantity changing at the instant when the ...

Calculus- Related rates
A spherical balloon is inflated at the rate of 1 cm^3 per minuter. At the instant when the radius r=1.5, (a.) how fast is the radius increasing? (b.) how fast is the surface area increasing?

Calculus
In 1979, a biologist Reto Zach published a study on how crows drop whelks, a type of mollusk, from a height that minimized the amount of energy spent to break open the shells. Drop from too low a height, and the bird has to pick the shell up many times before it breaks. Drop ...

Calculus
Quotient Rule: Use the limit definition of the derivative to prove that the quotient rule

calculus
it is assumed that f is differentiable and that w has an absolute maximum at t0 w(t)= f(t)/(c+t) derivative is f'(t)(c+t)-f(t)/(c+t)^2 Show that f(t0) = f′(t0)(C + t0). I'm having a bit of trouble in the above question. I keep getting f(t0)= (c+t0)-f'(t0) instead of f(t0) = ...

Calculus
A child is flying a kite. If the kite is 135 feet above the child's hand level and the wind is blowing it on a horizontal course at 7 feet per second, the child is paying out cord at ______ feet per second when 285 feet of cord are out. Assume that the cord remains straight ...

calculus
w(t)= f(t)/(c+t) w'(t)=? I got f'(t)(c+t)-f(t)/(c+t)^2 as the derivative and I'm having a hard time trying to prove that f(t_0) = f'(t_0) (C+t_0) Help is always appreciated :)

calculus
The base of a certain solid is the triangle with vertices at (−6,3), (3,3), and the origin. Cross-sections perpendicular to the y-axis are squares. Then the volume of the solid?

Calculus
The graph of the equation is x^2+xy+y^2=9 a) What is the equation of the right most vertical tangent? b) That tangent touches the ellipse where y= what? I've calculated the derivative to y'=(-y-2x)/(2y+x) and I found the horizontal tangents. How do I do this part?

Calculus
Consider the given function and the given interval. f(x) = (x − 3)^2, [2, 5] (a) Find the average value fave of f on the given interval. fave = (b) Find c such that fave = f(c). c = (smaller value) c= (larger value)

Pre Calculus
Am I correct? Please show me how if I am wrong 1.How many 6 digit numbers can be made using each of the digits {8,8,9,5,5,6} exactly once? 6!=702 2.How many more arrangements are there from all of the letters of the word MARMALADE than there are from all of the letters of the ...

calculus
Find the area of the region between the curves y = x^2 and y = 2/(x^2+1).

calculus
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then ﬁnd the area of the region. y = 5x^2 and y = x^2+6

calculus
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. 2y = 4√x and y = 5 and 2y+2x = 6.

Calculus
A road perpendicular to a highway leads to a farmhouse located 6 mile away. An automobile traveling on the highway passes through this intersection at a speed of 75mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 10 miles...

pre calculus
A projectile is fired straight upward from ground level with an initial velocity of 320 feet per second. (Assume t = 0 seconds corresponds to the time the object is fired. Use 32 feet/second2 as acceleration due to gravity.) (a) At what instant will it be back at ground level...

Calculus
A street light is at the top of a 14.5 ft. tall pole. A man 5.3 ft tall walks away from the pole with a speed of 5.5 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 47 feet from the pole?

Pre Calculus 30
1.How many positive integers five-digit integers end with the digit 0? 9x10x10x10x? what number would represent zero 2.Using the digits {1,2,5,6,7,9} and not allowing repetition of digits, how many positive three-digit integers can be made that are larger than 500? 9x5x4 3....

calculus
find y' if x^2y^4+3y-4x^3=5cos x-1 (2x*y^4)+dy/dx 4y^3*x^2)+dy/dx3+12x^2=5-sin-0 dy/dx=-5sin (x -12 x^2)/(2x*4y^3)+(12y^2+x^2) this is using implicit differenation is this correct

calculus
find y' if x^2y^4+3y-4x^3=5 cos x-1 (2x-y^4)+(dy/dx4y^3*x^) +dy/dx+12x^2=5(-sin x-1) is this started correctly

calculus
Chocolate Box Company is going to make open-topped boxes out of 7 × 11-inch rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way? (Round your answer to one decimal place.)

Calculus
Suppose the integral from 2 to 6 of g of x, dx equals 12 and the integral from 5 to 6 of g of x, dx equals negative 3 , find the value of the integral from 2 to 5 of 3 times g of x, dx

calculus
How would you use series to evaluate the lim(x->0) of (x-arctanx)/x^3? I'm getting stuck, and the answer says it's 1/3 but I don't know how they got there.

calculus
How would you determine the power series of 1/(1-x)^3. I know that the series of 1/(1-x) is x^n, but how would you manipulate it for this scenario?

Calculus
The length l of a rectangle is decreasing at the rate of 3cm/sec, while its width w is increasing at the rate of 3cm/sec. Find the rates of change of (1.) the area, (2.) the perimeter, (3.) the length of one diagonal at the instant when l=15 and w=6.

Calculus
A point is moving along the curve xy=12. When the point is at (4,3), the x-coordinate decreases at the rate of 2cm/sec. How fast is the y-coordinate changing at that point?

Trig identies, Calculus
Use the identities cos^2 x + sin^2 x =1 and cos2x=cos^2 x -sin^2 x to show that cos^4 x -sin^4 x = cos2x Im not sure how, I can solve my problem with half angle identities but im not sure where to start with this.

Calculus
Find the derivative of the integral from x^3 to 5 of cosk/(k^2+2). My answer is ((-cos(x^3)(3x^2))/(x^6+2).

Calculus
Find the derivative of the integral from x to 6 of ln(1 + t^2). My answer is -ln(1+x^2).

Calculus II
If {an} (a sequence) is decreasing and an > 0 for all n, then {an} is convergent. True/False?

Pre Calculus 30
1.How many positive four-digit integers can be formed that are even?9x9x9x5=3645 2.Three differently colored six-sided dice are rolled. In how many different ways can the dice turn up? 6x6x6=216 3.How many positive integers five-digit integers start with the digit 1?1x9x9x9x9=...

Calculus
Evaluate the integral: the integral of the cube root of x, dx. a. (3/4)x^(4/3)+C b. (3/2)x^(2/3)+C c. 1/(3x^(2/3))+C d. None of these I got (3x^(4/3)/4)+C. I was wondering if it would be D.

Calculus
Evaluate the integral. 1/2 integral e^(t/2) (I'm not sure what the 1/2 on the left of the integral symbol means.)

Calculus
An explosion causes debris to rise vertically with an initial velocity of 160 feet per second. In how many seconds does it attain maximum height? i know i need to use the formula: y(t)=-16t^2+Vot + yo Have: vo=160 Yo= is what i am trying to find and i dont know how to solve ...

Math/Calculus
Find the equation of the tangent line to the given equation at the indicated point a) y^2=cos(5x-3y) at ((2pi+3)/5, 1) b) x^2+x^2y+4=0 at (2,-2)

Calculus
Kayla claims that she can find an approximate value for sin 1o without using a machine to do any computations. Explain or show how she can do this (Hint: if you choose to explain the process, please be very specific. If you choose to show how it is done, you can use your ...

Differential equations,Calculus
So I have the following differential equation. The general solution I have is: t=k(-1/r)+c I now need to find the particular solution when t=0 and the radius (r) = 1cm. So k is a constant which is approx 3.9 (5/4pi) So for the particular solution im really just plugging in the...

Calculus (RATE OF CHANGE)
Two cars leave a town at the same time and travel at constant speeds along straight roads that meet at an angle of 60° in the town. If one car travels twice as fast as the other and the distance between them increases at the rate of 81 mi/h, how fast is the slower car ...

Calculus (RATE OF CHANGE)
Estimate the instantaneous rate of change at the point indicated. (Round your answer to four decimal places.) y(x) = 1/ x + 2; x = 2

Calculus
What is y=-2/7x-1 in standard form?

Calculus
What is y=-18x+13 in standard form?

calculus
One litre of water at 0 degrees Celsius will have a different volume at higher temperatures. For a temperature 0 ≤ T ≤ 30 (in Celsius) this volume (in litres) is well approximated by the function V = −aT 3 + bT 2 − cT + 1 using the coefficients a = 9.8 × 10−8, b = 3...

Calculus
Integrate sqrt(16+x^2) dx

Calculus
Integrate x^2/(sqrt(x^2-25)dx

Math - Calculus
Directions (or bearings) on Earth are measured in degrees, running from zero to 360 degrees, clockwise, starting with 0 degrees being due North. So due East for example, is 90 degrees, due South 180 degrees, and Northwest is 315 degrees. You are swinging a rock clockwise (...

Math
The letters in the words ALGEBRA, GEOMETRY, and CALCULUS are represented by the sets V= {A, L, G, E, B, R}, W= {G, E, O, M, E, T, R, Y}, and X={C, A, L, U, S}, respectively. I don't understand this can someone help?

Differential equation, calculus
Hiya, Dt/dr =k/r^2 Is this directly solvable or do I need to separate variables. I've got t=-k/r+c from the former and something along the lines of r^3=kt+c from the latter. Think I'm doing it all wrong. Thanks

Pre Calculus 30
1.How many positive three-digit integers can be made from the digits {3,4,5,6,7} if digits may be repeated? 5x4x3=60 2.In how many ways can all of the letters of the word HEXAGON be arranged if the arrangement must start with a consonant and end in a vowel? 4x3x1x2x3x4x5=1440 ...

calculus
A convex lens of a focal length f can be defined by the lens equation 1/f=1/p + 1/q if an object is a distance p from the lens, then the distance q from the lens to the images. For a particular lens, f=2cm and p is increasing find 1. a general formula for the rate of change of...

calculus
The position of a particle is given by the equation of motion s(t)= 1/(1+t)where t is in sec and s in meters find the velocity.

Pre Calculus 30
I'm having a hard time understanding the fundamental counting principal and I don't even know where to start on some of the questions.Here are a few questions I'm having trouble with. 1.Andrea, Brian, Carol, David, Emmalee, Floyd, and Gloria are to stand in a line for a ...

Calculus
Please help me solve these problems. 1.Find the inverse function (f^-1) of √3x-2 2. Integration ∫ the top is x while the bottom is 1. (sin(t))/t dt

Calculus II
Find the centroid (x¯,y¯)(x¯,y¯) of the region bounded by: y=5x^2+6x y=0 x=0 x=6 I know the integral (think, at least) is from 0 to 6 and between the first and second equation given, but I missed this lecture and have no idea how to find the center of mass

Pre-Calculus
How do I solve the equation: 3 = 2.1 sin[(pi/6)t - 5pi/6] + 3.2

calculus
The voltage in a electric current is 100 volts then with Ohm's law how do you find the instantaneous rate of change of I with respect to R ta a resistance of 20 ohms if R is increasing I know how to use the instantaneous formula but what is f(a)

Calculus
Given k(p) = |p|+4, find the average value of this function on the interval from -4 to 4. I get 6.

Math - Calculus
f(x) = { x^2sin(1/x), if x =/= 0 0, if x=0} a. find lim(x->0)f(x) and show that f(x) is continuous at x=0. b, find f'(0) using the definition of the derivative at x=0: f'(x)=lim(x->0) (f(x)-f(0)/x) c. Show that lim(x->0)f'(x) does not exist. In particular, this means ...

Calculus
Find the derivative of the integral from 2 to 5x of (sqrt(1 + u^2))/u. My answer: (sqrt(1 + 25x^2)/5x) * 5﻿﻿﻿﻿

Calculus Question! ASAP!
Hello! I have this problem: x(dx)/sqrt(9-x^2) I was wondering why I can't use trig substitution and substitute sqrt(9-x^2) for sqrt(1-sec^2) and having: integral x = 3sin(theta) dx = 3cos(theta)d(theata) integral 3sin(theta)(3cos(theta))/3cos(theta) having the 3cos(theta) ...

IB Calculus
A particle moves along the x-axis so that at any time t≥0, its velocity is given by v(t)=t^2-16t+4 What is the velocity of the particle when its acceleration is zero?

Calculus
Air is being pumped into a spherical hot air balloon at a rate of 50 cm^3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 200 cm.

Pre calculus
find 2 exponential functions with asymptote of y=80 , with points (0,176) (1,169) (3,156) (5,146) (20,101) (30,90) (60,81).one should have the base of "e"

pre-Calculus
The coffee starts out at 176 degrees, and cools off like this… Min Degrees 1 169 2 162 3 156 5 146 10 125 15 111 20 101 30 90 60 81 If he measured the coffee at 2 hours and 3 hours, what temperature would it be?

Math - Calculus
The horizontal velocity is constant. (Ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed v in a direction that makes an angle α with the horizontal, then the initial vertical velocity vv and the horizontal velocity vh are ...

Calculus
M = log(E)-11.4/1.5 How do you express E in terms of M?

Pre calculus
explain how to do these with please. I was out a couple of days so give clear instructions. 1. sin(π/2-ϴ)secϴ 2. cot(π/2-ϴ)cosϴ

Pre-Calculus
explain how to do these with please. I was out a couple of days so give clear instructions. sin(π/2-ϴ)secϴ cot(π/2-x)cosϴ

Calculus
A hang glider is standing at the top of a 3,000 foot cliff. The hang glider jumps off and begins to descend at a constant rate of 50 feet per second, how fast is the area of the triangle formed by the cliff, the hang glider, and the ground changing at the instant when the hang...

calculus
can someone also help me ndertsand this question ; y=x^3 +2x^2+3 represents the rate of an object over the interval [7, 10]. What is the average rate of the object over the time interval?

calculus
can someone help me with this problem y=x^2 -9 represents the velocity of an object over the interval [1, 5] in meters per second. What is the displacement of the object over the time interval?

Calculus
Differentiate each 1. Y= 1/x + 3/x^2 + 2/x^3 2. Y= x^5(1+ x)^5 3. Y= 4-x^2 / 2x+3 Can someone help me I'm lost

Simplify the cubed root of 432x to the 12th power

Math; Calculus
You toss a rock up vertically at an initial speed of 55 feet per second and release it at an initial height of 6 feet. The rock will remain in the air for_______ seconds. It will reach a maximum height of_______ feet after ________ seconds.

College calculus
Find the dy/dx A. y = u sqrt u + 1; u = 2x^2 - 2/3 B. x = u/ (1 + u^3); y = u^2 /(1 + u^3) Can someone help me

Calculus
A point moves along the curve y= x^3 -3x + 5 so that x= 1/2 square root t + 3 where t is the time. At what rate is y changing when t = 4? Need some help!!!

basic calculus
A page is to contain 54 square centimeters of printed material. If the margins are 1 cm at the top and bottom and 0.5 cm at the sides, find the most economical width of the page (in cm).

Calculus
Show that there are at least two values of x in the interval (−π,π) such that x=1+sin(x)