1. the partial derivative of f(x,y) = x^2y^x with respect to x is x^3y^(x-1) Tru
ID: 2857915 • Letter: 1
Question
1. the partial derivative of f(x,y) = x^2y^x with respect to x is x^3y^(x-1)
True or False
2. The global maximum of a function f(x,y) on a domain D in the plane always occurs at a point (a,b) where the partial derivatives of f vanish.
True or False
3. There is a function f(x,y) such that both partial derivatives of f vanish at (0,0) but that point is neither a local max nor a local min.
True or False
4. A vector has three components: direction, magnitude, and position
True or False
5. Any line in R^3 is determined by an equation of the form ax + by + cz = d
True or False
6. Suppose f(x) is a continuous function of one real variable. Define F(x) = integral from x to 0 f(y)dy. The fundamental theorem of calculus states in part that
a. the derivative of F with respect to x is f
b. the derivative of f with respect to x is F
7. You are riding home from class on a skateboard and want to figure out how far your journey is, taking into account the precise path you travel (including turns, hills, and vert tricks). What technique is most appropriate for this calculation?
a.Integration of vector-valued functions
b.Differentiation of vector-valued functions
c.Double integration in polar coordinates.
d. Taking the cross product of the curvature and integrating.
Explanation / Answer
1) f(x) = x2yx
partially differentiating with respect to x (Assume y is a constant)
==> f/x = (2x2-1)yx + x2 yx lnx since (uv)' = u'v + uv' , d/dx ax = ax lna
==> f/x = (2x)yx + x2 yx lnx
==> f/x = xyx (2 + xlnx)
Hence False
2) False . since the point where partial derivatives vanish is either local maximum or local minimum or a saddle
point. It is not the global maximum of the function.
3) True . Since the point could be a saddle point. partial derivatives vanish not only at maximum or minimum but also saddle points.
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