Water is running out of a conical funnel at the rate of 1 cubic inch per second. If the radius of the top of the funnel is 4 inches and the height is 8 inches, find the rate at which the water level is dropping when it is 2 inches from the top.
The curve y = x^3 + x^2 - x has two horizontal tangents. Find the distance between these two tangents and draw a picture. Please show how to solve the following problem to the answer of 32/27.
If u = x2 +y2 and v = 2xy, transform F(u,v) into G(x,y). Find δG/δx and δG/δx in terms of δF/δu and δG/δy. Please see the attached file for the fully formatted problem.
Y=(x+1)/(x^2-4) 1) Find dy/dx and second derivative 2) What are the asymptotes? 3) Intervals of increase? intervals of upward concavity? Points where the horizontal line is tangent?
A balloon rises at a rate of 3 meters per second from a point 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observor when the balloon is 30 meters above the ground. Answer: 1/20 radian per second
When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume V satisfy the equation PV^1.3 = k where k is a constant. Find the relationship between the related rates dp/dt and dV/dt Answer: V^0.3 (1.3p dV/dt) + V dp/dt = 0
A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above ground . When he is 10 feet from the base of the light, (a) at what rate is the tip of his shadow moving? (b) at what rate is the length of his shadow changing? Answer: 25/3 feet per sec 10/3 feet per sec
A baseball diamond has the shape of a square with sides 90 feet long. A player running from second base to third base at a speed of 28 feet per second is 30 feet from third base. At what rate is the player's distance s from home plate changing?
A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat. (a) the winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to dock? (b) Suppose the boat is moving at a constant r
Please use differentiation and related rates to solve the following problems. Please explain answers and solve to specified solutions. A Ladder is 25 feet long and is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top moving down
At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately 3x the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?
The included angle of the two sides of constant equal length s of an isosceles triangle is Z degrees. (a) Show that the area of the triangle is given by A = 1/2s^2 sin Z (b) If Z is increasing at the rate of 1/2 radian per minute, find the rate of change of the area when Z = pi/6 and Z = pi/3 (c) Explain why the rat
Please verify each differentiation formula and explain how. (a) d/dx[arctan u] = u'/1 + u^2 (b) d/dx[arcsec u] = u'/lul (square root of u^2 -1) (c) d/dx[arccos u] = -u'/square root of 1 - u^2 (d) d/dx[arccot u] = -u'/1- u^2 (e) d/dx[arccsc u] = -u'/lul (square root of u^2 - 1
Please explain how to find dy/dx using logarithmic differentiation. Please solve to the designated answer. y = x to the 2/x power Answer: 2(1 - ln x)x^(2/x - 2) power
Please explain how to find d^2y/dx^2 in terms of x and y. Please solve to the designated answer. y^2 = x^3 Answer: 3x/4y
Please explain how to find the equations of the tangent line and the normal line to the graph of the equation at the indicated point and achieve the specified answer. y ln x + y^2 = 0 (e, -1)
Find dy/dx using logarithmic differentiation (a) y = x √(x^2 - 1) Answer: 2x^2 - 1/√(x^2 - 1) (b) y = x^2(√(3x - 2)/(x - 1)^2 Answer 3x^3 - 15x^2 + 8x/2(x - 1)^3 √(3x - 2) (c) y = x^2/x Answer : 2(1-ln x)x^2/x-2
Find the slope of the tangent line at the indicated point. (a) Witch of Agnessi (x^2 + 4)y = 8 Point: (2, 1) (b) Bifolium (x^2 + y^2)^2 = 4(x^2)y Point: (1,1)
Find dy/dx by implicit differentiation and evaluate the derivative at the indicated point. xy = 4 at the point (-4,-1)
Let u be a differentiable function of x. Use the fact that lul = the square root of u^2 to prove that d/dx[lul] = u'(u/lul). u is not equal to zero.
Air is being pumped into a spherical balloon so that the radius is increasing at the rate of dr/dt = 3 inches per second. What is the rate of change of the volume of the balloon in cubic inches per second, when r = 8 inches? Hint: V = 4/3(pi)r^3
Please explain how to solve to the following problem: A buoy oscillates in simple harmonic motion y = A cos omega(t) The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its
How do I find the anti derivative of the attached problem?
2 (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1m/s how fast is the area of the spill increasing when the radius
(See attached file for full problem description with equations) --- real numbers complex numbers Suppose , are continuous functions with nonvanishing first partial derivatives. Let , Show that . ---
Differentiate: 1) y=(cos(x)^cot^2(x) 2) find the second derivative of sin(x^2*y)=-cos(x^2*y) note: *=multiply
The curve y = ax2(ax sqaured) + bx passes through the point (2, 4) with gradient 8. Find a and b. I have the answer but do not understand how to get to it. Therefore, could you please show full workings.
The function has a derivative everywhere and has just one critical point, . In parts (a)-(d), you are given additional conditions. In each case decide whether is a local maximum, a local minimum, or neither. Explain your reasoning. Sketch possible graphs for all four cases. a) b) c) d) Please see th
Derivative of a functions, slope of a curve. See attached files for full problem description.
Assume that you collect P dollars from a transaction and being a mathematics wiz, you have developed formula to calculate the future value of your investment: where, r is the rate of interest and t is the time horizon. Suppose you invest your profit, P dollars, from above transaction, and invest it in a bank at 5% rate of