44 Answered Questions for the topic Multivariable Calculus

Multivariable Calculus

11/08/21

Letf=x2 +y2(1+x)3 (a) Show that f has only one critical point and that it is a local minimum at that critical point. (b) Show that f has no minimum value.
Let f = x^2 +y^2 (1+x)^3(a) Show that f has only one critical point and that it is a local minimum at that critical point.(b) Show that f has no minimum value.
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Multivariable Calculus

10/25/21

Explain your answers in the following.
(a) Show that the minimum of x+y+z subject to xyz = 1,x ≥ 0,y ≥ 0,z ≥ 0is3. Be sure to show why the minimum exists. Suggestion: Use the gradients to find the critical point, on the constraint... more
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Multivariable Calculus

10/25/21

(a) For a constant a, find the average y value for the region between y = a + 1 and y = x2 + a. (b) For what value of a is the average y value in Part (a) zero?
(a) For a constant a, find the average y value for the region between y = a + 1 and y = x^2 + a.(b) For what value of a is the average y value in Part (a) zero?
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Multivariable Calculus Calculus

09/14/21

Find the length of the shortest curve between (1, 0, 0) and (1, 1, 1) on the surface x2 + y2 = z2 + 1. Be sure to explain why your answer is correct.

Find the length of the shortest curve between (1, 0, 0) and (1, 1, 1) on the surface x2 + y2 = z2 + 1. Be sure to explain why your answer is correct.
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Multivariable Calculus Calculus

09/14/21

Consider the dashed line through the (0,0,0) and (0,1,1), and the dotted line through (0, 1, 0) and (1, 0, 0) as in the figure.
Consider the dashed line through the (0,0,0) and (0,1,1), and the dotted line through (0, 1, 0) and (1, 0, 0) as in the figure.(a) Let f(t) be the point that moves along the dashed line, with... more
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Multivariable Calculus Calculus

09/14/21

Find the equation of the plane through the point (1, 2, 3) such that it contains the line in the z = 0 plane given by x + y = 1.
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Multivariable Calculus Math

07/01/21

Find the domain and range of f(x,y) = (xy + y²) / (x² + y²) over the real numbers, and find the limit as (x,y) --> (0,0) of f(x,y):
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Multivariable Calculus Math Calculus Mathematics

01/03/21

Approximating a multivariable plane using a set of single variable equations
I am attempting to make a multivariable equation that describes pump flow when 2 different characteristics are manipulated. My equations are:y = 8.1429x + 0.0057 when z=10y = 7.3571x + 0.0043 when... more
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11/11/20

How to find double integral with change of variables
I am trying to complete a double integral with change of variables, and am not sure where I am going wrong (somewhere in setting up the integral is my guess). Thank you for any help you can... more
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Multivariable Calculus Math Integral Calculus

09/17/20

Evaluate the line integral with respect to arc length.3 variables
Evaluate the line integral with respect to arc length where C is given by ∫c x ds x = t, y = t sin t, z = t cos t, 0 ≤ t ≤ √(2)firstly subbed t into equation as x=tthen differentiated x, y and... more
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Multivariable Calculus Directional Derivatives

05/29/20

Please help with this problem.

You are to design a stainless-steel fermenter, which has a right circular cylindrical shape with a height of 1.2 m and radius of 0.3 m. How fast will the fermenter’s volume change with a small... more
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Multivariable Calculus Calculus

03/24/20

Multivariable Calculus Question - Global max & min
Find the global maximum and minimum to the function f (x,y,z) = xy2z3 on the set S = (x, y, z) : x2 + y2 + z2 ≤ 1 .
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Multivariable Calculus Calculus Vector Calculus

01/11/20

Projectile Motion using Vector Calculus
Trebuchet can throw an 80 lb rock up to 100 yards. The maximum height that the rock reaches as it flies through the air toward a fortress wall 100 yards away is 20 yards, calculate the initial... more
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Multivariable Calculus Gradient Tangent Orthogonal

01/03/20

Find the plane which is tangent to a surface at a point.
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Multivariable Calculus Math Calculus Algebra

06/01/19

Multivariable Calculus Problem
Please help me with the problem below, I am really stuck:In 2-space, find the unit vectors orthogonal to y = e ^ 2x at x = 0.
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Multivariable Calculus Gre Reference Request Gre Exam

04/04/19

Math GRE: Calculus Textbooks - is Spivak + Stewart + Rudin sufficient?
Recently, I splurged and spent $1000 in math textbooks in preparation for the Mathematics GRE subject test. So far, in terms of calculus books, I have purchased Spivak, Stewart, and Baby Rudin.... more
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Multivariable Calculus Calculus Gre Gre Exam

03/24/19

GRE Mathematics Multivariable Calculus?
I'm studying for the GRE subject test on 4/5, and I've been going through the princeton review book, which seems to be the canonical guide. In the multivariable section the theorems are pretty... more
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Multivariable Calculus Calculus Real Analysis Complex Analysis

03/22/19

Geometrical Interpretation of Cauchy Riemann equations?
Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation. Do the Cauchy Riemann equations have a geometric interpretation?
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Multivariable Calculus Linear Algebra

03/18/19

Divergence as transpose of gradient?
In his online lectures on Computational Science, Prof. Gilbert Strang often interprets divergence as the "transpose" of the gradient, for example [here](http://youtu.be/gYME3EbIqV4?t=31m50s) (at... more
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Multivariable Calculus Calculus Optimization Lagrange Multiplier

03/15/19

How to know whether Lagrange multipliers gives maximum or minimum?
My book tells me that of the solutions to the Lagrange system, the smallest is the minimum of the function given the constraint and the largest is the maximum given that one actually exists. But... more
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Multivariable Calculus Calculus Optimization

03/14/19

Why does a distance and its square reach their minimum at the same point?
There is a question in my calculus textbook that asks to find a point on the parabola $y^2 = 2x$ that is closest to point $(1,4)$. They want us to first use the distance formula, but then... more
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Multivariable Calculus Triple Integrals

02/01/19

Evaluating triple integral with tetrahedron
∫∫∫xyz dV where T is is the solid tetrahedron with vertices (0,0,0) (1,0,0) (1,1,0) (1,0,1). How do I figure out what z=f(x,y) is with these points? The equation of a tetrahedron is... more
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Multivariable Calculus Chain Rule Calculus Proofs

08/18/18

Chain law with functions of several variables
A function 'f' of two variables is said to be homogeneous of degree 'n' if f(tx,ty) = t^n*f(x,y) whenever t > 0.   How can I show that such a function 'f' satisfies the equation: x*(partial... more
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Multivariable Calculus

04/15/18

Converting Triple Integrals to Spherical Coordinates.
X goes from 0 to 2 Y goes from 0 to sqrt(4-x^2) Z goes from 0 to sqrt(4-x^2-y^2).   The function is zsqrt(4-x^2-y^2)dzdydx.   I know the conversion from rectangular to spherical and the... more
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Multivariable Calculus

06/26/17

Find the minimum distance from the origin to the surface
Find the minimum distance from the origin to the surface z2 = (x-1)2+(y-1)2   
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