# Calculus

Calculus
A rotating light is located 25 feet from a wall. The light projected on the wall is moving at a rate of 1.5 feet per second when the light's angle is 15 degrees from perpendicular to the wall. If the light is turning at a constant rate, how many seconds does it take to go ...

calculus
A rotating light is located 25 feet from a wall. The light projected on the wall is moving at a rate of 1.5 feet per second when the light's angle is 15 degrees from perpendicular to the wall. If the light is turning at a constant rate, how many seconds does it take to go ...

Calculus
A rotating light is located 25 feet from a wall. The light projected on the wall is moving at a rate of 1.5 feet per second when the light's angle is 15 degrees from perpendicular to the wall. If the light is turning at a constant rate, how many seconds does it take to go ...

Math (Calculus II) (Weird Work Problem Setup Help)
I am trying to understand my teacher's example of a Work problem. Cut to the chase here's a picture of the problem: goo.gl/photos/AqZ6ENmHLg6heixD7 While I understand how to integrate fairly well, I'm still confused on how exactly my teacher set up the work problem. Since work...

multivariate calculus
Q.NO.1: Show that the function z=ln(x^2 + y^2)+2tan^-1(y/x) satisfies Laplace’s equation. (∂^2z/∂x^2)+(∂^2z/∂y^2)=0

Calculus
Find the limit. lim 1/x^2-4 a)x->2^+ b)x->2^- c)x->-2^+ d)x->-2^- I'm completely lost. I don't understand how some answers are infinity and the others are negative infinity.

The rate at which water flows into a tank, in gallons per hour, is given by a differentiable function R of time t. The table below gives the rate as measured at various times in an 8-hour time period. t---------0-----2------3-------7----8 (hours) R(t)--1.95---2.5---2.8----4.00...

Calculus
f is a continuous function with a domain [−3, 9] such that f of x equals 3 for x between negative 3 and 0 including negative 3, equals negative 1 times x plus 3 for x between 0 and 6 inclusive, and equals negative 3 for x greater than 6 and less than or equal to 9 and let g ...

calculus
Q.NO.4: Find the arc length of the graph of r(t). r(t)=(t^2)i +(cost + t sint)j +(sint- t cost)k , 0≤t≤π

calculus
Q.NO.3: Show that the function z=tan^-1(2xy/x^2-y^2)satisfies Laplace’s equation; then make the substitution x=r cosθ, y= r sinθ and show that the resulting function of satisfies the polar form of laplace’s equation (∂^2z/∂r^2)+(1/r^2)(∂^2z/∂θ^2)+​(1/r...

calculus
Q.N0.2: Show that u(x,y)=ln(x^2 + y^2) and v(x,y)=2tan^-1(y/x) satisfy Cauchy-Riemann equations (∂u/∂x)=(∂v/∂y) and (∂u/∂y)=(-∂v/∂x)

calculus
Q.NO.1: Show that the function z=ln(x^2 + y^2)+2tan^-1(y/x) satisfies Laplace’s equation. (∂^2z/∂x^2)+(∂^2z/∂y^2)=0

Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 4 of x squared, dx.

calculus
Elasticity a) Find the elasticity of the demand function q + 2p = 5000 when p = \$1000, and q = 3000. b) How would revenue be affected by a price increase?

Calculus I'm really confused
solve the differential equation dy/dx=y^2/x^3 for y=f(x) with condition y(1)=1.

The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of integral from 1 to 2 f(x) dx =.1*4/2+.1*7/2+.3*10/2+.2*13/2+.2(15/2)+​.1*18/2 =4.40

Calculus
The slope of the tangent to a curve at any point (x, y) on the curve is -x/y . Find the equation of the curve if the point (3,-4) on the curve.

Calculus
Suppose an airline policy states that all the baggage must be boxed shaped with a sum of length, width, and height not exceeding 138 inches. What are the dimensions and volume of a square based box with the greatest volume under these conditions.

Calculus
A light in a lighthouse 5 kilometers offshore from a straight shoreline is rotating at 4 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point 5 kilometers from the point opposite the​ lighthouse?

calculus
Find the area of the region enclosed by the parametric equation x = t^3−3t y = 6t^2.

calculus
Notice that the curve given by the parametric equations x=25−t^2 y=t^3−16t is symmetric about the x-axis. (If t gives us the point (x,y),then −t will give (x,−y)). At which x value is the tangent to this curve horizontal? x = ? At which t value is the tangent to this ...

calculus
Find the length of the curve deﬁned by the parametric equations x = 2/3t, y = 2ln((t/3)^2−1) from t =6 to t =7.

Calculus
A rectangular page is to contain 8 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the ...

Calculus
At what point is the function y=csc(2x) continuous?

calculus
from a cardboard box 12 in by 8 inches are cut out so the sides can be folded up to make a box . What dimentions will yield a maximum volume? What is maximum volume? we got our V = 4x^3--40x^2+96x V'=12x^2-80x +96 then we know to use quadratic equation I get 20( +/-) sqrt 112/...

calculus
If x = 18cos^3θ and y = 18sin^3θ, ﬁnd the total length of the curve swept out by the point (x,y) as θ ranges from 0 to 2π.

calculus
Consider the curve deﬁned by the equation y = 4x^3 +3x. Set up an integral that represents the length of curve from the point (0,0) to the point (4,268).

Calculus
Compute the absolute and relative errors in using x to approximate x. x=pi; c=3.18

Calculus
Consider the differential equation dy/dx = 2x - y. Let y = f(x) be the particular solution to the differential equation with the initial condition f(2) = 3. Does f have a relative min, relative max, or neither at x = 2? Since we're trying to find a max/min I don't think it's ...

Calculus
If lim x->4 f(x)-5/x-2 =1, find lim x->4 f(x). I know the answer is 7, I need help on solving it. Thanks

Pre-calculus
Can someone help me with this equation: y = (2*10^(-44))*(e^(0.055x))?

calculus
Hi, I have the following HW question and I'm a little confused: Find the volume of the solid generated by revolving the region bounded by y=1/sqrt(x) and y=0 for 1 <= x <= 2 about the line y=-1. So far I have the integral 2pi (-1-y)(1-1/y^2) dy from -1 to 1, but that's ...

calculus
Find the area of the shaded region. r^2=sin(2θ).

calculus
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r=6/θ, θ=π

calculus
Find a Cartesian equation for the curve and identify it. (r^2)cos(2θ)=1

calculus
Find the exact area of the surface obtained by rotating the given curve about the x-axis. x=t^3, y=t^2, 0 ≤ t ≤ 1

calculus
Find the area enclosed by the curve x = (t^2)-2t, y = (t)^(1/2) and the y-axis.

Calculus
I am given the true of false question: If f(x) is continuous and 0<=f(x)<=1 for all x in the interval [0,1], then for some number x f(x)=x. This seems intuitively true, but I'm not sure. All help is greatly appreciated.

Calculus
A particle's position along the x-axis is given by x(t) = t^4/24 - t^3/2 + 2t^2 - 1. What is the maximum acceleration on the interval 0 <= t <= 4? I found the acceleration to be (t^2 - 6t + 8)/2 and the derivative of the acceleration to be t - 3. I thought the maximum ...

Calculus
A differentiable function called f(x) achieves its maximum when x=0. Which of the following must then be true? 1. The function p(x) = xf(x) has a critical point when x = 0. 2. The function m(x) = (f(x))^2 has its maximum at x = 0. 3. The function j(x) = f(x^2) has its maximum ...

Calculus
Let the length of a rod be 10 meters and the linear density of the rod ρ(x) be written in the form ρ(x) = ax + b with x = 0 representing the left end of the rod and x = 10 representing the right end of the rod. If the density of the rod is 2kg/m at the left end and 17 kg/m ...

calculus
Hi, could someone please help me with this hw question asap? Given that the integral of (e^x*sin(5x))dx=((e^x)/26)*(sin(5x)-5cos​(5x))+c, evaluate the integral from 1 to e^(pi/10) of sin(5ln(x))dx

calculus
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=tcos(t), y=tsin(t); t=π

calculus
x = sin(θ/2), y= cos(θ/2), −π ≤ θ ≤ π. Eliminate the parameter to find a Cartesian equation of the curve.

Calculus
Use cylindrical shells to find the volume formed by rotating the region in the first quadrant enclosed by: y=1.2-1.4|x-12| and y=0 about the y-axis

Statistics
In how many different ways can a chemistry book, a calculus book, a history book and a dictionary be arranged on a shelf so that the chemistry book or the history book appears first?

Calculus
Use cylindrical shells to find the volume formed by rotating the region in the first quadrant enclosed by: y=1.2-1.4|x-12| and y=0 about the y-axis

Calculus
Use washers to find the volume formed by rotating the region enclosed by: y=1.4−2|x−13| and y=0 about the y-axis

Calculus I
A particle on the x-axis is moving to the right at 2 units per second. A second particle is moving down the y-axis at the rate of 3 units per second. At a certain instant the first particle is at the point (5,0) and the second is at the point (0,7). How rapidly is the angle ...

An object has a constant acceleration of 30 ft/sec2, an initial velocity of −10 ft/sec, and an initial position of 4 ft. Find the position function, s(t), describing the motion of the object. s(t)=4-10t+15^2

Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = -1. ANSWER: v = ∫[1,e] π(4-(lnx+1)^2) dx

Calculus
Find the area of the region bounded by the curves of y=sin^-1(x/4), y=0, and x=4 obtained by integrating with respect to y. Your work must include the definite integral and the anti-derivative. I am really confused on this question. I graphed all of the points, but I have no ...

Calculus
Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x^3 - 9x on the interval [-1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem. I think one point is zero.

Calculus
The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x^2. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid? So I got 1/30 because (integral from 0 to 1) (x-x^2...

Math (Calculus) (Work)
Six and one-half foot-pounds of work is required to compress a spring 4 inches from its natural length. Find the work required to compress the spring an additional one-half inch. (Round your answer to two decimal places.) This is what I did so far: 4 in = 1/3 ft 6.5 = ∫[0,1/...

ASL
1. Nodding your head should be added to which statement or question? (1 point) Who is your brother? Recently, I took an American Sign Language course. Yes, I agree. No, I don’t think so. 2. Shaking your head should be added to which statement or question? (1 point) Who is ...

Calculus
Which of the following integrals will find the volume of the solid that is formed when the region bounded by the graphs of y = ex, x = 1, and y = 1 is revolved around the line y = -2

Calculus
For an object whose velocity in ft/sec is given by v(t) = −2t^2 + 4, what is its distance travelled, in feet, on the interval t = 0 to t = 2 secs? I think the answer is 2.667 but I am not one hundred percent sure.

Calculus
find the area of the region bounded by the graphs of y=x^2 and y=cos(x)

Calculus
find the area of the region bounded by the graphs of y=-x^2+5x+2 and y=2

calculus
Let y = sinx + ztanx,where z is a function of x. If (dy/dx)= z+ cosx, show that (d^2y/dx^2) =-y. I really have no idea how to do this. Please show me

Calculus
Find the distance traveled by a particle with position (x, y) as t varies in the given time interval. x = cos^2 t, y = cos t, 0 ≤ t ≤ 9π and the length of the curve so far i got ∫0 to 9pi √((-2cos(t)sin(t))^2+(-sint)^2) I am having trouble integrating from here

Calculus
Determine the remainder for Sn where n=5 or n=9 for the following series. ∑n=1 to ∞ (n^2+1)/(n^4) S5= R5= S9= R9=

Calculus
find the volume of the solid formed by revolving the region bounded by the graphs of y=x^3 y=1 x=2 about the y-axis

Calculus
The temperature of a pan of hot water varies according to Newton's Law of Cooling: dT dt equals negative k times the quantity T minus A, where T is the water temperature, A is the room temperature, and k is a positive constant. If the water cools from 90Â°C to 85Â°C in 1 ...

Calculus
The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y. What is the equation of the curve if (2, 1) is a point on the curve?

Math (Calculus II) (Volume Revolution Setup)
Consider the graph of y^2 = x(4-x)^2 (see link). Find the volumes of the solids that are generated when the loop of this graph is revolved about (a) the x-axis, (b) the y-axis, and (c) the line x = 4. goo.gl/photos/v5qJLDztqsZpHR9d7 I'm just having trouble trying to set up the...

Calculus
Find the first 5 partial sums of the series ∑n=2 to ∞ 1/(n(n+4))? I got 1/12 for the first one but when I put 1/21 for the next one it says it's wrong please help

Calculus
The divergence test applied to the series ∑n=1 to ∞ 3n/(8n+9) tells us that the series converges or diverges? I got that it was divergent because it was undefined at infinity, is my answer right?

calculus
The differential equation below models the temperature of a 87°C cup of coffee in a 17°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 67°C. Solve the differential equation to find an expression for the temperature of the...

Hello! I am struggling with this problem: Find the Taylor Series for sin(x-2) centered at c=3. My work so far: sin(1)+(x-3)cos(1)-(1/2)(x-3)^2sin(1)-(1​/6)(x-3)^3cos(1)+(1/24)(x-3)^4sin(1)... I know that the Taylor series for sign is typically: the summation from n=0 to ...

Water is poured into a bucket according to the rate F(t)=(t+7)/(t+2) , and at the same time empties out through a hole in the bottom at the rate E(t)=(ln(t+4))/t+2 , with both F(t) and E(t) measured in pints per minute. How much water, to the nearest pint, is in the bucket at ...

Calculus
Find the area of the region bounded by y = x^2, y = 0, x = -1, and x = 2. I tried the integral from -1 to 2 of x^2 and got 3 as the answer. I tried (integral from 0 to 1 of √y + 1) + (integral from 0 to 4 of 2 - √y) and got 13/3. What is wrong with the way the integrals ...

The velocity of a particle moving along the x-axis is v(t) = t^2 + 2t + 1, with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance traveled by the particle from t = 0 to t = 2 minutes.

Pre-Calculus
Annual sales of fountain pens in Littleville are 2,000 per year and are increasing by 10% per year. How many fountain pens will be sold over the next five years? Use an integral to solve this problem. I just need help setting the problem up - thank you!

Calculus

Calculus
The radius of a circle is increasing at a rate of 3 ft/min. If the radius is 0 when the time is 0, how fast is the circle's area changing when time is at 10 minutes?

Math - Calculus
Suppose you live near a bay where the water level fluctuates due to the tides. Your bay is an inverted cone with a radius of 1 mile, and a depth in the center of 100 feet. (There are 5280 feet in a mile.) Water flows through a channel in and out of the bay with the tides. You ...

Math(Calculus)
Im having a real difficult time solving this question. Find the absolute extrema of the function. h(x) = e^x^(2) - 4 on [-2,2] Absolute maximum value: at x = Absolute minimum value: at x =

Calculus
Use the disk method to find the volume of the solid generated by revolving about the y-axis the region bounded by the curves y=8−x^2 and the curve y=x^2.

Calculus - Fundamental thm of Calc
Use the Fundamental Theorem of Calculus to find the derivative of f(x)=∫[4,x^2]((1/4)t^2−1)^15 dt

Calculus - Related Rates
A road perpendicular to a highway leads to a farmhouse located 5 miles away. An automobile traveling on the highway passes through this intersection at a speed of 65mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 1 miles...

Calculus Help
Suppose you drill a circular hole with radius r through the center of a sphere with radius R. You remove exactly half the volume of the sphere. The ratio of your radii is: r/R=

Find the range of the function F(x)= definite integral from [-6,x] of sqrt(36-t^2)dt. [0, 36π] [0, 18π]*** [-6, 6] [-6, 0]

Pre calculus
write the finite series -1+2+7+14+23+...+62 in summation notation

Pre calculus
When “superballs” sprang upon the scene in the 1960’s, kids across the United States were amazed that these hard rubber balls could bounce to 90% of the height from which they were dropped. If a superball is dropped from a height of 2 m, how far does it travel by the ...

Calculus
The function f is continuous on the closed interval [-5,5], and f(-2) = 6, f(1) = -3, and f(4) = 6. Which of the following statements must be true? A. The equation f(x) = 0 has at least two solutions on the closed interval [-5,5]. B. The equation f(x) = 0 has exactly two ...

calculus
find the parametric equation of the line that is tangent to the parabola y=x^2 at the point(-2 , 4)

Calculus
If dy/dt = k/y and k is a nonzero constant, which of the following could be y? A. y = √(2kt + 16) B. y = kt + 5 C. y = √(kt + 16) D. y = 5e^(kt) E. y = √(2kt) + 4 I got E by finding the integral of y dy = k dt.

calculus
Find the exact area of the surface obtained by rotating the curve about the x-axis. y=((x^3)/4)+(1/3X), 1/2≤X≤1

Calculus (trig derivatives)
A lighthouse 2 miles offshore has a light that rotates once every 20 seconds. At what rate is the light traveling along the shore if it is shining on a point 4 miles along the shore from the point nearest the lighthouse?

Calculus
If g(x) = integral from 0 to x^2 of √(t^3+2) then g'(2) = ?

Calculus
If y = 18 - 3x, what is the maximum value of the product 2xy? A. -3 B. 0 C. 3 D. 6 E. 54 I found that the value of the maximum occurs at x = 3, but that the value itself is 54. Is E the answer?

calculus
In this problem, you will investigate the error in the nth degree Taylor approximation to ln(x+1) about 0 for various values of n. (a)Let E_1=ln(x+1)−P_1(x)=ln(x+1)−(x). Using a calculator or computer, graph E1 for −0.1≤x≤0.1, and notice what shape the graph is. Then...

Calculus PLSSSSS HELP DUE SOON
Assuming P≥0, suppose that a population develops according to the logistic equation dP/dt=0.03P−0.00015P^2 where tt is measured in weeks. Answer the following questions. 1. What is the carrying capacity? I tried solving the differential equation and got 600 but it says it'...

Calculus 1 Help
Find the volume generated by rotating about the given line a) y^2=x, x=2y; about the y-axis b) y=x, y=sqrtx; about y=1

Calculus Math
Find the height of y=((sqrtx)(3-x))/3 for 0<=x<=3 Use the arc length formula and note that 1/4x+1/2+x/4=(1/(2sqrtx) + sqrtx/x)^2 is a perfect suare.

Calculus Help
A solid generated by rotating about the x-axis the region under the curve of f(x) from x=0 to x=b, is b^2 FOR ALL b>0. Find the function f.

Math - Calculus
Find the volume of the solid given by rotating the region bounded by the curves y=x^2, x=1, x=2, and y=0 around the y-axis a) Use the shell method b) Use the washer method. Be careful with the radius of the washer at different y.