- Some Questions On The Gradient 1 Suppose 2 V In The Temperature In C Of A Fiat Sheet Metal At Position X Y In C 1 (32.35 KiB) Viewed 36 times
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- Some Questions On The Gradient 1 Suppose 2 V In The Temperature In C Of A Fiat Sheet Metal At Position X Y In C 4 (29.25 KiB) Viewed 36 times
Some questions on the gradient.
(1) Suppose f (x, y) is the temperature (in ◦C) of a flat sheet of metal at position (x, y) (in cm). Suppose
∇f (7, 2) = (-2, 4)
Suppose an ant walks on the pan. It’s position (in cm) at time t (in s) is given by ~r (t). We have
~r (6) = (7, 2)
and
~r (6) = (-3, 4)
By “the temperature of the ant,” we mean the temperature at the position of the ant.
(a) What are the units of ∇f?
(b) How would you interpret ~r (6) = (-3, 4) within this problem? Answer using a sentence about
the ant. Include units in your answer.
(c) What is the instantaneous rate of change of the temperature of the ant per second of time, at
time t = 6 s? Include units in your answer.
(d) What is the instantaneous rate of change of the temperature of the ant per centimeter the ant
travels, at time t = 6 s? Include units in your answer.
(e) Standing at the point (7, 2), in which direction should the the ant walk so it’s instantaneous
rate of change of temperature will be as rapid as possible? Give your answer as a unit vector.
(f) If the ant at (7, 2) walks in the direction given by (e), what will be the instantaneous rate at
which the ant warms up per cm travelled at that moment? Include units in your answer.
(g) If the ant at (7, 2) walks in the direction given by (e) at a rate of 3 cm/s, what will be the
instantaneous rate at which the ant warms up per second at that moment? Include units in
Some questions on the gradient (1) Suppose (2.v) in the temperature in "C) of a fiat sheet metal at position (x,y) (in cm). Suppone V/7,2)=(-2,4) Suppose an ant wall on the pan. It's position (in cm) attimet (in) to given by 7(e). We have F(6) (7.2) (6)-(-3.4 By the temperature of the ant" wo mean the temperature at the position of the att (a) What are the units of Vf? and (b) How would you interpret 7" (6) -(-3.4) within this problem? Answer wing a sentence about theant. Include it in your we (c) What is the instantaneous rate of change of the temperature of the sat per second of time, at time=6s? Inchade units in your answer (d) What is the instantaneous rate of change of the temperature of the ant per centimeter the est travels, attimet Include it in your we (e) Standing the point (7.2), in which direction should the theatwalk so it's anos rate of change of temperature will be as rapides pole? Give your wat vector () If the ant at (7.2) walles in the direction given by (e. what will be the stantaneous atest which the antwurms upper en travelled at that moment includes your (1) If the stat (7.3) walk in the direction given by ste of what will be the torte at which the person that he cale in
Some questions on the gradient, (1) Suppose } (x,) is the temperature (in °C) of a flat sheet of metal at position (7.) (in cm). Suppose V/(7.2) - (-2,4) Suppose an ant walks on the pan. It's position (in em) at time t (in s) is given by (6). We have (6) (7.2) and (6) -(-3,4) By the temperature of the ant," we mean the temperature at the position of the ant. (a) What are the units of 1 (b) How would you interpret (6) - (-3, 4) within this problem? Answer using a sentence about the ant. Include units in your answer. (©) What is the instantaneous rate of change of the temperature of the ant per second of time, at timet - 6x? Include units in your answer. (d) What is the instantaneous rate of change of the temperature of the ant per centimeter the ant travels, at time t = 6s? Include units in your answer.
(e) Standing at the point (7,2), in which direction should the the ant walk so it's instantaneous rate of change of temperature will be as rapid as possible? Give your answer as a unit vector. (f) If the ant at (7,2) walks in the direction given by (e), what will be the instantaneous rate at which the ant warms up per em travelled at that moment? Include units in your answer. (g) If the ant at (7,2) walks in the direction given by (e) at a rate of 3 cm/s, what will be the instantaneous rate at which the ant warms up per second at that moment? Include units in your answer.