Calculus Questions - HELP

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In summary, the conversation revolves around six different calculus questions. The questions cover topics such as finding coordinates and equations of spheres, planes, and surfaces, proving geometric relationships, and determining maximum values of functions subject to constraints. The specific questions ask for the coordinates and center of a cube, equations of spheres contained within and between the cube, an equation of a plane that intersects another plane at a 60 degree angle, equations of tangent planes and normal lines to given surfaces, and values of constants for a critical point and type of critical point. Lastly, the conversation also mentions the need for a homework section and requests for any work done on the problems.
  • #1
jherasjr
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Calculus Questions - HELP!

1. Consider the cube determined by the planes x=-1, x=3, y=5, y=9,
z=0, and z=4.

a) Give the coordinates of the eight vertices and center of
the cube.

b) Determine an equation of the largest sphere contained in
the cube.

c) Determine an equation of the largest sphere that would fit
between the sphere found in (b) and the cube.

2. Determine an equation of a plane that intersects the plane
x+y+z=3 at an angle of 60 degrees.

3. a) Determine an equation of the tangent plane to the surface
given by x^2*y+y^2*z+z^2*x=1 at the point (1,0,1).

b) Determine an equation of the line that is normal to the
surface given by x^2*y+y^2*z+z^2*x=1 at the point (1,0,1).

4. Prove that u+v+w=0, then u*v=v*w, and u*w=w*v. What is the
geometric interpretation of these relationships?

5. Suppose f(x,y)=A*X^3+B*X*Y+C*Y^2, where A, B, and C are
constants. For what values of A, B, and C does f have a
critical value at (-2,1)? Determine what type of critical
point it is.

6. Determine the maximum value of f(x,y,z)=(x*y*z)^2 subject to
the constraint x^2+y^2+z^2=c^2, where c is not equal to zero.
 
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  • #2
Why is this not in the homework section?

Presuming they are homework, show us what you have done on these problems.
 
  • #3


1. a) The eight vertices of the cube are (-1,5,0), (-1,5,4), (-1,9,0), (-1,9,4), (3,5,0), (3,5,4), (3,9,0), (3,9,4). The center of the cube is (1,7,2).

b) The largest sphere contained in the cube has a radius of 2 and is centered at (1,7,2). Therefore, its equation is (x-1)^2 + (y-7)^2 + (z-2)^2 = 4.

c) The largest sphere that would fit between the sphere found in (b) and the cube would have a radius of 1 and be centered at (1,7,2). Therefore, its equation is (x-1)^2 + (y-7)^2 + (z-2)^2 = 1.

2. To find the equation of a plane that intersects x+y+z=3 at an angle of 60 degrees, we need to find a vector that is perpendicular to the plane. We can use the cross product of two vectors in the plane, for example (1,0,1) and (1,-1,0) to find this perpendicular vector. This gives us the vector (1,-1,-1). The equation of the plane can then be written as x-y-z=d, where d is a constant. To find d, we can plug in the coordinates of the point of intersection (1,1,1) into the equation, giving us d=3. Therefore, the equation of the plane is x-y-z=3.

3. a) To find the equation of the tangent plane at the point (1,0,1), we can use the gradient of the surface given by x^2*y+y^2*z+z^2*x=1. The gradient is ∇f = (2xy+z^2, x^2+2yz, 2zx+y^2). Plugging in the coordinates of the point (1,0,1), we get ∇f(1,0,1) = (1,1,1). Therefore, the equation of the tangent plane is x+y+z=3.

b) The normal vector to the surface at the point (1,0,1
 

1. What is calculus and why is it important?

Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. It is important because it provides a framework for understanding and analyzing natural phenomena such as motion, growth, and change in various fields including physics, engineering, and economics.

2. What are the two main branches of calculus?

The two main branches of calculus are differential calculus and integral calculus. Differential calculus deals with the study of rates of change, while integral calculus deals with the study of accumulation and the inverse process of differentiation.

3. What are the basic concepts in calculus?

The basic concepts in calculus include derivatives, integrals, limits, and functions. Derivatives are used to find the rate of change of a function, while integrals are used to find the area under a curve. Limits are used to define the behavior of a function near a specific point, and functions are mathematical expressions that relate one variable to another.

4. How is calculus used in real life?

Calculus has numerous real-life applications, including predicting the motion of objects, determining maximum and minimum values, optimizing systems, and analyzing economic trends. It is also used in fields such as engineering, physics, and finance to model and solve complex problems.

5. What are some tips for success in learning calculus?

Some tips for success in learning calculus include practicing regularly, seeking help when needed, breaking down problems into smaller steps, understanding the underlying concepts, and applying the concepts to real-world scenarios. It is also helpful to review algebra and trigonometry concepts before diving into calculus.

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