# Calculus

Calculus
A hot air balloon rising vertically is tracked by an observer located 2 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is π5 , and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at...

Calculus
A car travels down a highway at 45 m/s. An observer stands 250 m from the highway. a) How fast is the distance from the observer to the car increasing when the car passes in front of the observer? (b) How fast is the distance increasing 30 s later?

Calculus
A man of height 1.4 meters walk away from a 5-meter lamppost at a speed of 2.9 m/s. Find the rate at which his shadow is increasing in length.

Calculus
A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 7 m from the dock?

Calculus
A 18 ft ladder leans against a wall. The bottom of the ladder is 3 ft from the wall at time t=0 and slides away from the wall at a rate of 2ft/sec. Find the velocity of the top of the ladder at time t=2

Calculus
A balloon rises vertically at a rate of 8 feet/sec. A bird flies 40 feet above ground toward the balloon’s path at 20 feet per second. At what rate is the distance between the bird and the balloon changing when the bird is 50 feet from the balloon (straight line distance) ...

Calculus
If there is a critical point between intervals where a function is increasing on both intervals, is there a relative max/min at that point?

Calculus
Find the intervals on which x^2/(x-3)^2 is decreasing. a. (0,6) b. (0,3), (3,6) c. (-inf,0), (6,inf) The critical numbers are x=0 and 6, but since there is an asymptote at x=3, I'm not sure whether a or b is the answer.

Calculus: Optimization
A person on a lake in a canoe 1 mile from the nearest point "P" on a straight shore line; the person wishes to get to a point "Q" , 10 miles along the shore from "P". To do so, the canoe moves to a point "R" between P and Q and then walks the remaining distance to"Q". The ...

Pre-Calculus
Solve the equation on the interval [0,2pi). 2sin^2x-3sinx+1=0 (2sinx+1)(sinx+1) I don't think I did the factoring correctly. When I multiply it out to double check I get 2sin^2x+3sinx+1

Calculus
Consider the area between the graphs x+2y=4 and x+4=y2. This area can be computed in two different ways using integrals 1) Compute as a sum of two integrals 2) Compute as a single integral 3) Either way, what is the area? __ So for 1) I rewrote the two to be: x = 4-2y x = y^2-...

calculus 1
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x) = 4x3 + x2 + 4x

calculus
One ship is sailing South at the rate of 5 knots, and another is sailing East at a rate of 10 knots. At 2 p.m the second ship was at the place occupied by the first ship one hour before. At what time was the distance between the ships not changing?

Calculus
Find dy/dx for y=3tan^-1(x/2). I know the answer is 6/x^2+4 but I'm stuck at y'=3(1/1+(x/2)^2)(1/2)

Calculus
A piece of ice stays in the shape of a sphere as it melts. The volume of the ice is decreasing at a constant rate of 3π/2 cubic feet per hour. What is the rate of change of the surface area of the piece of ice when the radius is 10 feet? I found the rate of change of the ...

Math Calculus
Sketch the graph of a function having the following features: f'(x)>0 on (-5,-2) and on (3,oo) f'(x)<0 on (-oo,-5) and on (-2,3) f''(x)>0 on (-oo,-3) and on (1,oo) f''(x)<0 on (-3,1) Please help!

calculus
A water trough is 8m long and its cross-section is an isosceles triangle which is 60cm wide at the top, and the height is 60cm. The trough is not full. Give an expression for V, the volume of water in the trough in cm^3, when the depth of the water is d cm. Thank you in advance!!

Calculus 1
At 2:00 pm a car's speedomoter reads 20 mph, and at 2:10 pm it reads 25 mph. Use the Mean Value Theorem to find an acceleration the car must achieve.

Calculus II
Hi, I can't seem to find out how to do this. My textbook has two Q(t) values in the example and I don't know how to interpret this problem. An institute finds that the average student taking Elementary Machine Shorthand will progress at a rate given by dQ/dt = k(85 − Q) in a...

BC Calculus
Fluid is flowing in a tube that has a radius of 3 centimeters. Water is flowing through a circular cross section at a rate of (9-r^2) cm/s, where r is the distance from the center of the cross section. What is the total amount of water that flows through the cross section in 4...

Calculus
describe the transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3 give the order in which they must be preformed to obtain the correct graph pls help!!!

pre calculus
Double Angle Formula. If the double angle formula for cos2(u) = 2cos^2(u)-1, then write the double angle formula for cos2(u) in terms of sine.

pre calculus
how long will it take for an investment to triple if it is compounded continuously at 4.2%

calculus
At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 24 knots and ship B is sailing north at 16 knots. How fast (in knots) is the distance between the ships changing at 3 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Calculus
Suppose the height of an object fired straight up from the level ground is given by s(t)=300t-4.9t where s is measured in meters and t is measured in seconds. how fast is the object moving up as it leaves the ground after 2 seconds? find the average velocity on the interval [1...

Calculus
a pebble is dropped into a calm pond, causing ripples in the form of concentric circles. the radius r of the outer ripple is increasing at a constant rate of 1 root per second. When the radius is 4 feet, at what rate is the total area A of a disturbed water changing? Thank you!

Calculus
A stone is dropped into the sea, causing circular waves whose radii increases at a constant rate of 1.5 per minute. At what rate is the circumference of the wave is changing? Thank you!

Calculus
Water is flowing into a vertical cylindrical tank at the rate of 24 cu. ft. per minute. If the radius of the tank is 4 feet, how fast is the surface rising? Thank You! :)

Calculus
A pole 12 meters long is leaning against a vertical wall of a building. if the foot of the pole is being pulled away from the wall at the rate of 2 meter per minute, how fast is the top of the pole sliding down the wall at the instant when the foot of the pole is 5 meters from...

calculus
Differentiate the function.((x^2)-((3x^2)(ln(4x))))/(x^6)

Calculus 3
The base of an decorative rectangular feature with a volume of V = 225 ft^3 is to be constructed in a restaurant. The bottom is made of marble, the sides are made of glass, and the top is open. If marble costs five times as much (per unit area) as glass, find the dimensions of...

Calculus II
I'm having trouble getting started. Steve said it's similar to another question I asked but I don't know how to begin. Any help? Suppose a cell of volume V cc is surrounded by a homogeneous chemical solution of concentration C g/cc. Let y denote the concentration of the solute...

Calculus II
Suppose a cell of volume V cc is surrounded by a homogeneous chemical solution of concentration C g/cc. Let y denote the concentration of the solute inside the cell at any time t and suppose that, initially, the concentration is y0. Fick's law, named after the German ...

Calculus II
Assume that the rate of change of the unit price of a commodity is propirtional to the difference between the demand and the supply, so that dp/dt = k(D-S) where k is a constant of proportionality. Suppose that D = 72-5p, S = 9 + 2P, and p(0) = 2. Find a formula for p(t).

Pre-Calculus

Calculus
Given that f(150)=100, f'(150)= -2.5, approximate the value of f(151.5). The answer is approximately 96.25, but I'm unsure of how my teacher has gotten that answer. Can someone please explain this to me?

Calculus
A is the doubling time for y=26e^0.5t, and B is the doubling time for y=13e^0.5t. Then (a) A=2B (b) B=2A (c) A=B (d) A=(ln2)B The correct answer is A, right? I just want to make sure I got the correct answer.

Calculus 1
A filter filled with liquid is in the shape of a vertex-down cone with a height of 12 inches and a diameter of 18 inches at its open (upper) end. If the liquid drips out the bottom of the filter at the constant rate of 5 cubic inches per second, how fast is the level of the ...

calculus
3cotx - radical 3 = 0 solve for x please show it step by step

Pre-Calculus
When buying gasoline, you notice that 14 gallons of gasoline is approximately the same as 53 liters. Use this information to find a linear model that relates gallons to liters. Then use the model to find the number of liters in 25 gallons. Round your answer to the nearest tenth.

Pre-Calculus
If w varies directly as z and w = 9 when z = -6, find w when z = 3. a) 0 b) -4.5 c) -2 d) 4.5

Pre-Calculus
True or false: Suppose that y varies directly as x. If y is 5 when x is 30, the constant of variation is 150.

Pre-Calculus
True or false: P varies directly as Q can be written as P = kQ.

Calculus
1) If y = 2x3 - 4x and dx, dt equals 4 , find dy/ dt when x = 1. 2) The area A = πr^2 of a circular water ripple changes with the radius. At what rate does the area change with respect to the radius when r = 4ft? 3) A rectangular box has a square base with an edge length of x...

Calculus
1) A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(2t). Find the acceleration of the particle when the velocity is first zero. 2) The driver of a car traveling at 50 ft/sec suddenly applies the brakes. The position of the car is s...

AP Calculus
- Which of the following is false for f(x) = (5x^3-5x^2-10x)/(2x^5-2x) a) The x-axis is an asymptote of f(x) b) x = -1 is not an asymptote of f(x). c) x = 1 is an asymptote of f(x). d) The y-axis is an asymptote of f(x). Thanks

Calculus
I have three questions I need help solving 1) A ball's position, in meters, as it travels every second is represented by the position function s(t) = 4.9t2 + 550. What is the velocity of the ball after 3 seconds? 2) The cost in dollars of producing x units of a particular ...

Calculus
1. For f(x) = (1-x)/(1+x) and g(x) = x/(1-x), find the simplified form for f[g(x)] and state the domain? 2. Evaluate lim x -->1 ((√x^2 +8) -3))/(x-1) 3. A particle moves on a line away from its initial position so that after t hours it is s = 4t^2+t miles from its initial...

Calculus
1. For f(x) = (1-x)/(1+x) and g(x) = x/(1-x), find the simplified form for f[g(x)] and state the domain? 2. Evaluate lim x -->1 ((√x^2 +8) -3))/(x-1) 3. A particle moves on a line away from its initial position so that after t hours it is s = 4t^2+t miles from its initial...

pre-calculus math
A polynomial f(x) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f(x) as a product of linear and/or quadratic polynomials with real coefficients that are irreducible over . 3, −3 − 2i;    degree 3

Calculus
A ball is thrown from a height of 130 feet with an initial downward velocity of 6/fts . The ball's height h (in feet) after t seconds is given by the following. =h−130−6t16t2 How long after the ball is thrown does it hit the ground?

Pre-Calculus
Verify the identity. (csc(2x) - sin(2x))/cot(2x)=cos(2x) =csc(2x)/cot(2x) - sin(2x)/cot(2x) =csc(2x)/cot(2x) - cos(2x) Is this correct so far? If so then how would I continue? I got stuck on this part...

Pre-Calculus
Verify the identity. cos x - sec x = -sin x tan x So far I have applied a reciprocal identity =cos x - 1/cos x I know I must combine using the least common denominator but I don't know how it would be written.

Pre-calculus
How do I factor this: 6m^2 + 7mn + n^2 I really don't understand this one

Pre-calculus
I have a really basic question. Say I have a rational expression and my numerator is 4xy. Am I allowed to factor out just the x or just the y? For example y(4x)?

Pre-calculus
Hi, How do I simplify and state the restrictions on the following: 12x^8y^9 / 3x^4y^6 what do I need to factor out? I'm so confused, I never know what I'm allowed to factor out.

PHILOSOPHY
1. Some regular churchgoers believe that taxpayers' dollars should not be used to fund laboratories that carry out tests on animals for medical research. Hence, it is wrong to go on spending taxpayers' dollars for that purpose. Answer: False Cause 2. Yogi Berra, an Italian ...

Calculus
A rectangle has a vertex on the line 3x + 4y = 12. Two of its sides lie along the positive x and y axes. Find the maximum possible area of such a rectangle.

Calculus
Use the given graph of f to find a number δ such that if 0 < |x − 3| < δ then |f(x) − 2| < 0.5. since i cannot put up the graph ill put the coordinates (2.6,1.5) (3,2) (3.8,2.5) Please help im not sure how to solve this

calculus
Find an equation of the tangent line to the graph at the given point. (The graph is called a Witch of Agnesi.) f(x)= 7 / x^2 + 3 (2,1)

Calculus-Math
Delta Cephei is one of the most visible stars in the night sky. Its brightness has periods of 5.4 days, the average brightness is 4.0 and its brightness varies by plus-or-minus 0.35. Find a formula that models the brightness, b, of Delta Cephei as a function of time, t, with t...

Pre Calculus
Find all $2 \times 2$ matrices $\bold{A}$ that have the property that for any $2 \times 2$ matrix $\bold{B}$, $\bold{A} \bold{B} = \bold{B} \bold{A}.$

Pre Calculus
For a vector $\bold{v}$, let $\bold{r}$ be the reflection of $\bold{v}$ over the line $\bold{x} = t \begin{pmatrix} 2 \\ -1 \end{pmatrix}.$ [asy] unitsize(1 cm); pair O, V, R; O = (0,0); V = (3,1); R = reflect(O,(2,-1))*(V); xaxis(-2,4); yaxis(-3,1); draw((-1)*(2,-1)--2*(2,-...

Pre Calculus
Let $\bold{w} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. There exists a $2 \times 2$ matrix $\bold{P}$ such that $\text{proj}_{\bold{w}} \bold{v} = \bold{P} \bold{v}$ for all 2-dimensional vectors $\bold{v}$. Find $\bold{P}$.

Pre Calculus
Find the 2 x 2 matrix A such that pmatrix{(-1,-1)(1 -1)} A = I, where I is the 2 x 2 identity matrix.

Calculus
I really don't get this question :( What is 2+2? a. 3 b. 7 c. 4 d. -4 I really need help!!!

Math, calculus III
A particle moves along the path from the origin to (0, 2, 1) to (1, 2, 1) to (1, 0, 0) and back to the origin under the influence of the force field F = z2i + 2xyj+ 4y2k. Calculate the work done by direct calculation and by using Stokes’ Theorem

Math, calculus III
A fluid has constant density ρ and velocity v = zi + (y2 + 2y)j − 2xk. (a) Calculate the particle paths (field-lines). (b) Calculate the mass flux through the boundary of the region x2 + y2 ≤ 4 and 0 ≤ z ≤ (4−y) by direct calculation and by using the Divergence ...

math
Jill Moran has grades of 95 and 87 on her first two calculus tests. What is the minimum score on her third test that will give her an average of at least 90?

Calculus
You need \$12,000 in your account 5 years from now and the interest rate is 6% per year, compounded continuously. How much should you deposit now? Can someone explain the steps of solving this to me, please?

calculus
Differentiate the function. f(x) = e^7

Calculus
Rectilinear Motion Discuss the Velocity, Acceleration, and Speed: 1. s(t)=2t^2 +8t-7 2. s(t)=3t^4 -16t^3 +24t^2

Calculus
let ﻿f(x)=1x f(x)=1x ﻿and ﻿g(x)=x2 +5x.g(x)=x2 +5x. ﻿find ﻿(f⋅g)(x)(f⋅g)(x) Can someone help me with this I dont know how to do this

Calculus (Need Help) - Rectilinear Motion
Rectilinear Motion: *Need help with these three!* Directions: The position function of a particle moving on a coordinate line is given by the following eq'ns, where s is in feet and t is in sec. Describe the motion of the particle for any time. Make a DIAGRAM of the motion ...

Calculus
A mineral deposit along a strip of length 8 cm has density s(x)=0.07x(8−x) g/cm for 0≤x≤8. Calculate the total mass of the deposit. Your answer must include units.

Calculus
Calculate the population within a 6 mile radius of the city center if the radial population density is ρ(r) = 6(5+r2)1/3 (in thousands per square mile). (Use decimal notation. Give your answer to two decimal places.)

Calculus
Find the volume of the solid obtained by rotating the region enclosed by the curves y= 1/x and y= 6-x about the x-axis. (Use symbolic notation and fractions where needed.)

Calculus
Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = x^3 y = 16 x about the x-axis.

Calculus
Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 seconds. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function f(t)=1/2sin(2pi t/5) has often been used to ...

Pre calculus- help!
find (gof)(x) when f(x) = √x+1 and g(x) = x^2+3/x

calculus
if cos of theta= -4/5 and tan of theta > 0, find sin of theta

calculus
A person is jumping straight up and down on a trampoline. The height of the center of mass of the person is measured every tenth of a second. It takes just over one second to complete one full bounce. (A video of this situation is displayed above.) t (seconds) 2.90 3.00 3.10 3...

Question
@Alex asked this question when I was on earlier: Let line l_1 be the graph of 5x + 8y = -9. Line l_2 is perpendicular to line l_1 and passes through the point (10,10). If line l_2 is the graph of the equation y=mx +b, then find m+b. I'm wondering like how everyone did it and ...

Calculus
Suppose you throw an object from a great height, so that it reaches very nearly terminal velocity by time it hits the ground. By measuring the impact, you determine that this terminal velocity is -49 m/sec. A. Write the equation representing the velocity v(t) of the object at ...

Calculus (Volumes)
A solid has as its base the region bounded by the curves y = -2x^2 +2 and y = -x^2 +1. Find the volume of the solid if every cross section of a plane perpendicular to the x-axis is a trapezoid with lower base in the xy-plane, upper base equal to 1/2 the length of the lower ...

Calculus
a solid has as its base the region bounded by the curves y = -2x^2 +2 and y = -x^2 +1. Find the volume of the solid if every cross section of a plane perpendicular to the x-axis is a trapezoid with lower base in the xy-plane, upper base equal to 1/2 the length of the lower ...

calculus
h(t)=cot t a.) [pi/4,3pi/4] b.) [pi/6,pi/2] Find the average rate of change of the function over the give n interval and intervals. Please show step by step instructions. Thanks

Multiplevariable calculus
Hello, I have to find the volume of the region enclosed by the paraboloid z=(x^2)+(y^2) and the planes z=1 and z=3 , so well i am struggling about the limits of each integral. I know that I can use cylindrical coordinates, but what about the limits in z? I think that there ...

calculus II
A 100 gallon tank is being drained at a rate of 5 gallons per minute. A second tank containing 10 gallons of water is being filled at a rate of 4 gallons per minute. Can you find out when the two tanks will have the same amount of water in them?

Calculus
reflexive property of ellipse

Calculus
shiw that line tangents to an ellipse at the points P makes equal angle will the joining focus

Calculus
Suppose a certain object moves in a straight line with velocity v(t)= -2+t+3sin(pi t) where v is in meters per second and t is in seconds. Determine the net change in distance of the object from time t=0 to time t=6 seconds and find the object's average velocity on this ...

Calculus
Show that volume of the solid generated by revolving y=1/x between x=1,2x=infinity about x-axis

Calculus
Find the length of the parabola y^2=4ax cut off the line x=a about the x-axis

Calculus
The marginal cost for producing x units of a certain product is given by the formula MC = 0.001x^2-0.5x+66.5. A. What is the increase in the cost of the production if the production level is raised from 200 to 400 units? B. What is the average cost per item of producing 330 ...

pre-calculus
how to find the equation of a parabola given foci(-7,-17), (-7,17 ),the absolute value of the difference of the distances of any point from the foci is 24

Calculus II
Find the derivative of 5^(x^3+x) I know the answer, it's: 5^(x^3+x)*ln(5)*(3x^2+ 1) I just need some help on how to derive that. What rule is at work here? where did the ln come from?

calculus
Find the length of the curve given by the equation y= intergral from -pi to x of sqrt(cos(t)) dt for x between -pi and pi. I think I know to do this- at least part of it. I am using the fundamental theorem of calculus and the arc length formula, but I don't know how to find ...

Calculus
Hi, I am trying to find a closed form for the power series from n=2 to infinity of x^n/(n-1)n. I'm not really sure how to do this- I found the first and second derivatives which were x^(n-1)/(n-1) and x^n-2. However, this didnt really seem to help much. Any help is appreciated!

calculus
Use x=-1/2 in the MacLaurin series for e^x to approximate 1/sqrt (e) to four decimal places. So far I have |f^n(0)/n! (1/sqrt(e)-1/2)^n)| </= 1/10^4, but I don't know what to do from here.