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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_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_forwardAssume that the rate at which a cup of coffee cools to room temperature is proportional to the difference between its temperature T° C and the ambient room temperature R° C. A cup of coffee is placed in a 20°C room, cools from 100°C to 87°C in 3 minutes. The temperature varies over time t (in minutes) according to T = R+Aekt.t >0. a) Enter the values of R and A in the box below. R = Number A = Number b) Enter the exact value of k, in Maple syntax, in the box below. k %3D c) The temperature of the cup of coffee will be 60°C when t = Number Correct your answer to 1 decimal place. d) Enter the limiting value of its temperature in the box below. Numberarrow_forward
- The equation x²/3 + y²/3 = 4 describes an astroid: a. Find an expression for in terms of x and y. b. Find the equation of the tangent line to the astroid at the point (-3v3,1). c. There are 4 points on the astroid where the tangent line is not defined. What are these 4 points? What do you get if you evaluate at these points? dxarrow_forwardIn the next exercise you have to equal the derivative to zero and then clear x or a y to subtract in the given expression. That is, to form a system of equations. Given the implicit expression x* + y4 – 23 xy = 0, draw the graph and plot the three horizontal tangents with an appropriate domain. 1.arrow_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_forward
- A particle performs a movement on the same line from the origin. The equation that determines your distance in meters for a certain amount of time (in seconds) is d(t) = -0.5t3 + 4.5t? – 12t + 11 with t>0. The time in which the acceleration is zero corresponds to Note: Remember that velocity (speed) is the first derivative of distance and acceleration corresponds to the derivative of velocity (speed). a) t = 3 s b) t = 2 s c) t = 4 s d) t= 5 sarrow_forwardThe reaction of the body to a dose of medicine can sometimes be represented by an equation of the form R = M² where C is a positive constant and M is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, R is measured in millimeters of mercury. If the reaction is a change in temperature, R is measured in degrees, dR . This derivative, as a function of M, is called the sensitivity of the body to the medicine. dM and so on. Find dR dM = Ask my instructor F3 # 3 DII F4 4 8 F5 % 5 OL F6 A 6 F7 W & 7 8 F8 *** 8 F9 prt sc 9 F10 home ) F11 O Clear all end F12 insert + Check answer = 49 3:58 PM D 10/9/2022 delete backsparrow_forwardAnswer all the questions with CLEAR AND COMPLETE SOLUTIONSarrow_forward
- A particle of mass m moving through a fluid is is subjected to a viscous resistance R, which is a function of the velocity v. The relationship between the resistance R, velocity v, and time t is given by the equation: v(t) L R(u) m du. vo(t) Suppose that R(v) = -v/v for a particular fluid, where R is in newtons and v is in meters/second. If m = 10 kg and v(0) = 10 m/s, approximate the time required for the particle to slow to v = 5 m/s. %3Darrow_forwardGiven that a line has gradient- where a is a constant, find the gradient of another line perpendicular to it.arrow_forward
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