Problem 1:
Find dz/dy at the point (1,1,1) if z(x,y) is given implicitly by the equation
x^a[1]+y^a[2]+z^a[3]+a[5]*x*y*z^2 = 3+a[5]
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Problem 2:
Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ?
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Problem 3:
Find the directional derivative of the function f(x,y,z)
x^3*a[6]+y^3*a[3]+z^3*a[8]
At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3)
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Problem 4:
Find a saddle point of the function f(x,y)=
exp(x-a[4])-(x-a[4])*exp(y-a[6])
If there is no saddle point, write in the Answers: "Does Not exist". Explain what you are doing
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Problem 5:
Let f(x,y) be the function
a[4]*x + a[7]*y + a[2]
Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE
A = [a[1], a[2]], B = [a[3], a[4]], C = [a[5], a[6]]
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Problem 6:
Let f(x,y) be the function
(x^2*a[4]^2-y^2*a[5]^2)/(x*a[4]-y*a[5])
Find the LIMIT of f(x,y) as (x,y) goes to the point [a[5],a[4]], or show that it does not exist
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Problem 7:
Find the curvature of the curve
r(t) = [a[1], a[2]*t, a[3]*t^2]
At the point (a[1],0,0)
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Problem 8:
A particle is moving in the plane with ACCELERATION given by
[-a[1]*sin(t), -a[2]*cos(t)]
At time t=0 its position is , [0, a[2]]
and its velocity is , [a[1], 0]
Where is it located at time , t = Pi
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Problem 9:
A certain function depends on variables x and y
Right now the rate of change of the function with respect to x is, a[5]
and the rate of change of the function with respect to y is, a[7]
Both x and y depend on time
Right now the rate of change of x with respect to time is, a[1]
and the rate of change of y with respect to time is, a[9]
How fast is the function changing right now?
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Problem 10:
Find the point of intersection of the three planes
x = a[5], y = a[7], z = a[3]
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