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Derivatives

Derivatives : Speed of shadow - A tightrope is stretched 30 ft above the ground between Building 1(at point A) and Building 2( point B), which are 50 ft apart. A tightrope walker, walking at a ...

A tightrope is stretched 30 ft above the ground between Building 1(at point A) and Building 2( point B), which are 50 ft apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A. a) how far from point A is the tightrope walker when

Applications of Derivatives Word Problem

A plane flying with a constant speed of 24 km/min passes over a ground radar station at an altitude of 9 km and climbs at an angle of 40 degrees. At what rate is the distance from the plane to the radar station increasing 3 minutes later?

Rate of Change in Speed and Distance

(1) At noon, ship A is 60 nautical miles due west of ship B. Ship A is sailing west at 19 knots and ship B is sailing north at 25 knots. How fast is the distance between the ships changing at 8 PM, in knots? (2) A street light is at the top of a 22 ft tall pole. A woman 12 ft tall walks away from the pole with a speed of 9 f

Key data regarding differentiation

Explain whether each statement is true or false. Please give short explanation why and if false, please give an example. a) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval. b) If a function is continious on a closed interval, then it must have a minimum

Derivatives and Rate of Change : Rate of Increasing Area of an Isoceles Triangle

Let &#952; (theta) be the angle between equal sides of an isosceles triangle and let x be the length of these sides. If x is increasing at ½ meter per hour and &#952; (theta) is increasing pi/90 radians per hour, find the rate of increasing of the area when x=6 and &#952;=pi/4.

Derivatives: Tangents and Differentiation

Consider the relation defined by the equation tan y = x + y for x in the open interval 0 is less than or equal to x which is less than 2pi (a) Find dy/dx in terms of y (b) Find the x- and y- coordinate of each point where the tangent line to the graph is vertical (c) Find d^2y/dx^2 in terms of y

Derivatives

Please show how to solve to the answer of 1/2 If f(x) = x-1/x +1 for all x not equal to -1, then f'(1) =

Derivatives : Find Derivative from a Table of Values

Using the table of information for differentiable functions f(x) and g(x) at x = 2 and x=3, determine the derivative below. Chart: x f(x) g(x) f'(x) g'(x) 2 3 5 -2 -3 3

Derivatives and Rate of Change : Rate of Drop of Water Level in a Funnel

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.

Rate of Change in Angle of Elevation of a Balloon

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

Differentiation and Related Rates : Rate of Change of Length

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

Differentiation and Related Rates: Paseball Players Velocity

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?

Rate of Change of Speed of a Boat Being Pulled by a Winch

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

Differentiation and Related Rates

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

Rate of Change of Area of Isoceles Triangle

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

Verifying Differentiation of Trigonometric Functions : arctan, arcsec, arccos, arccot and arccsc

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

Application of Derivatives and Differentiation : Rate of Change of Volume of a Sphere ( Balloon )

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

Equation describing the motion of a buoy

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

Related Rates : Application of Derivatives Word Problems

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

Derivatives and Graphing : Local Maxima and Minima and Sketching Graphs

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

Application of Derivatives Word Problem : Future Value of Investment

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

Derivative

I am not sure how to set up this problem, I think that I have to use the exponential rule, but after that I am lost. (See attached file for full problem description with complete equations) --- The quantity q, of a certain skateboard sold depends on the selling price, p, in dollars, so, we write q = f(p). You are give

Applications of Derivatives : 10 Derivative Problems, Rate of Change, Pollution and Population Growth

1).Thermal Inversion When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 AM. Assume that the pollutant disperses horizontally, form

Set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints).

A). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints). Let p(x,y) = max_[0,1]|x'(t) - y'(t)|. 1).Show that ( M,p) fails to be a metric space. 2). Let p(x,y) = |x(0) - y(0)| + max_[0,1]|x'(t) - y'(t)|. Is (M,p) now a metric space? Please

Derivative Equation Constants

For this equation E= (-C/r) + D(-r/P) (Where c, D, and P are constants) Do the following procedure: 1. Differentiate E with respect to r and set the resulting expression equal to zero. 2. Solve for r0 in terms of C, D and P. Here is where I am at in the problem: I have obtained a derivative (and I'm look

Non Linear PDE Mathematical Symbols

I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of u with respect to x. Moreover, I will further simplify things by letting p=u*x and q=u*y. Also, ^ denotes a power (for example, x^2 means x squared) and / denotes division. This is the problem: T

Quotient and composite rule problem.

Please see attached problem using quotient and composite rule.

Solve the Derivative Problem

See the attached files. C(q) = 0.000002q^3 - o.o117q^2 + 84.446q + 23879 R(q) = -0.00003 * q^3 +0.0495q^2 + 118.02q P(q) = -0.000032q^3 + 0.0612q^2 + 33.554q - 23879 Use the Cost, Revenue, and Profit functions to find. a) C`(q) b) R`(q) c) P`(q) Do these equations predict the quantity needed to maximize profit, and th

Modelling the volume of a container/differentiation.

Question 1 A bucket-shaped container has a circular base of radius 10 cm, and its slant height is 30 cm. the radius of the open circular top of the container is 10x cm. the curved surface of the container is modeled by part of a cone, as shown below. Please see attached.

14 Derivative Problems : Product Rule, Quotient Rule, Chain Rule, First and Second Derivative and Finding Maximum or Minimum

Rules and Applications of the Derivative -------------------------------------------------------------------------------- 1. Use the Product Rule to find the derivatives of the following functions: a. f(X) = (1- X^2)*(1+100X) b. f(X) = (5X + X^-1)*(3X + X^2) c. f(X) = (X^.5)*(1-X) d. f(X) = (X^3 + X^4)*(30