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As in Problem 17, find the following gradients in two ways and show that your answers are equivalent.
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Mathematical Methods in the Physical Sciences
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- Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.arrow_forwardAn object with a mass of 10.00kg is launched from the ground with an initial upward velocity of 95.00m/s near the surface of Earth. Consider a linear model of the air resistance, kv, with v being the variable velocity and k = 0.0365 kg/s?. Take the ground as the origin and upwards as positive direction. 1. acceleration due to gravity. Which of the following best describes the DE relating to the problem? Let t be the time, g be the a. -10.00- 3.285 = 10.00 dv/dt b. -10.00g +0.0365v = 10.00 dv/dt c. -10.00g - 0.0365v = 10.00 dv/dt d. -10.00g - 3.285 = -10.00dv/dt 2 Which of the following is closest to the general solution of the linear DE model, velocity as a function of time? Let C be a real constant, exp()%3e0 a. v(t) = 1.346t + C b. v(t) = -2688+Cexp(t/274.0) c. v(t) = -2688+Cexp(-1/10.00) d. v(t) = -2688+Cexp(-1/274.0) %3Darrow_forwardPlease help me with 11.26 a.arrow_forward
- In a test of military ordnance, a large mumber of bombs were dropped on a target from various heights. The initial velocity of the bombs in the direction of the ground was 0. Let y be the height in meters from which a bomb is dropped, let x be the time in seconds for the bomb to strike the ground, let w = x, and let y= V. The relationship between x and y is given by y = 4.9x. For each of the following pairs of variables, state whether the correlation coefficient is an appropriate summary. a x andy b. w and y C x and v d. wand v e. Inxand Inyarrow_forwardA tire company has found that the quantity demanded x, in thousands of units per week, of their best-selling tire is related to the unit price p by the equation (see image). A.) Use differentials to find an approximate change in the quantity of tires demanded per week if the unit price of the tires is increased from $119 to $122 per tire. B.) Use your calculator to find the actual demand for tires when the price is set at $119 and $122 per tire. Then find the difference in demand. C.) Explain why the answers to parts A and B are not the same.arrow_forwardSuppose a rock falls from rest from a height of 100 meters and the only force acting on it is gravity. Find an equation for the velocity v(t) as a function of time, measured in meters per second. Hint What is the initial velocity of the rock?arrow_forward
- Find the gradient of ?=V x²+g?4Z? 24:arrow_forwardGiven the function z = x²y. a. Find the directional derivative of zat (1,4) in the direction making an angle -axis. Directional Derivative is b. Find the maximum rate of increase of zat (1,4). Maximum Rate of Increase is 3π 4 with the positive aarrow_forwardMatch the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you can probably solve this problem by guessing, it is useful to try to predict characteristics of the direction field and then match them to the picture. Here are some handy characteristics to start with -- you will develop more as you practice. A. Set y equal to zero and look at how the derivative behaves along the x-axis. B. Do the same for the y-axis by setting x equal to 0 C. Consider the curve in the plane defined by setting y = 0 -- this should correspond to the points in the picture where the slope is zero. D. Setting y equal to a constant other than zero gives the curve of points where the slope is that constant. These are called isoclines, and can be used to construct the direction field picture by hand. 1. y = 2y+ x²e2z 2. y = e + 2y 3. y = 2 sin(x) +1+y 4. y = y+2 A B Darrow_forward
- A 800-lb object is released from rest 600 ft above the ground and allowed to fall under the influence of gravity. Assuming that the force in pounds due to air resistance is - 20v, where v is the velocity of the object in ft/sec, determine the equation of motion of the object. When will the object hit the ground? Assume that the acceleration due to gravity is 32 ft / sec and let x(t) represent the distance the object has fallen in t seconds. Determine the equation of motion of the object. x(t) = Oarrow_forwardGiven that P = (10, 12), Q = (10.3, 12.1), ƒ(P) = 50 and f(Q) = 57, approximate the directional derivative of f in the direction from P to Q. NOTE: Round your answer to three decimal places. Directional derivative =arrow_forwardThe wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h=f(v,t), are recorded in feet in the following table. Use the table to find a linear approximation to the wave height function when v is near x knots and t is near 20 hours. Then estimate the wave heights when the wind has been blowing for 24 hours at y knots where x and y values given by above table. Wind speed (knots) V 20 t 30 40 50 60 5 5 9 14 19 24 10 7 13 21 29 37 Duration (hours) 15 8 16 25 36 47 20 8 17 28 40 54 30 9 18 31 45 62 40 9 19 33 48 67 50 9 19 33 50 69arrow_forward
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