Mathematics College

Compute the directional derivatives of the following functions along unit vectors at the indicated points in directions parallel to the given vector. (a) rx, y) = x^y, (Xo, yo) = (e, e), d = 8i + 15j (b) f(x, y, z) = e^x + yz, (Xo, yo, Zo) = (1, 1, 1), d = (1,-4, 4) (c) f(x, y, z) = xyz, (xo, yo, Zo) = (1, 0, 1), d = (1, 0,-1)

Answers

Answer 1

The derivative of given function is D(yz+e^x)d(1,1,1) = (3*((59)^1/2)*e)/59≈1.06167045295814

We have to find the directional derivative of the given function

f(x,y,z) = yz+e^x at (x0,y0,z0)=(1,1,1) in the direction of the vector

d⃗ =(3,−5,5)

Find the gradient of the function and evaluate it at the given point:

To find the gradient of a function (which is a vector), differentiate the function with respect to each variable.

∇f= (∂f/∂x,∂f/∂y,∂f/∂z)

∂f/∂x = ∂( yz+e^x )/∂x

The derivative of a sum/difference is the sum/difference of derivatives:

∂f/∂x = ∂( yz)/∂x +∂(e^x )/∂x since the derivative of y and z with respect of x is zero.

∂f/∂x = e^x The derivative of exponential is d(e^x)/dx=e^x.

Similarly

d(yz+e^x)/dy = z and d(yz+e^x)/dz = y

∇(yz+e^x)(x0,y0,z0)=(e^x,z,y)

Finally plug in the point we get the result

∇(yz+e^x)|(x0,y0,z0)=(1,1,1)=(e,1,1)

∇(yz+ex)|(x0,y0,z0)=(1,1,1)=(e,1,1)

Now we Find the length of the vector: |d⃗ |=((3)^2+(−5)^2+(5)^2)^1/2 =(59)^1/2

To normalize the vector, divide each component by length:

d = ((3*((59)^1/2))/59 ,(−5*(59)^1/2)/59,(5*(59)^1/2)/59)

Finally, the directional derivative is the dot product of the gradient and the normalized vector:

D(yz+e^x)d(1,1,1) = (e,1,1). ((3*((59)^1/2))/59 ,(−5*(59)^1/2)/59,(5*(59)^1/2)/59)

D(yz+e^x)d(1,1,1) = (3*((59)^1/2)*e)/59≈1.06167045295814

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Related Questions

the base of s is a circular disk with radius 5r. parallel cross-sections perpendicular to the base are squares.

Answers

The volume of the solid is 250r³ when the base of s is a circular disk with radius 5r.

Given that,

We have to discover the described solid's volume, v. The parallel cross-sections perpendicular to the base of s are squares, and the base of s is a circular disk with radius 5r.

We know that,

A cylinder is a solid with square cross sections parallel to the bases and circular bases. Its height is equal to the base circle's diameter.

The radius = 5r,

So the height = diameter = 10r

Volume of the cone is base × height= πr²×h

=π(5r)²×(10r)

=250πr³

Therefore, The volume of the solid is 250r³ when the base of s is a circular disk with radius 5r.

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Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(a, b) | a and b are the same age}
b) {(a, b) | a and b have the same parents}
c) {(a, b) | a and b share a common parent}
d) {(a, b) | a and b have met} e) {(a, b) | a and b speak a common language}

Answers

Relations in the options (a), (b) and (e) are the Equivalence relations as, the relations are reflexive, symmetric and transitive all.

Given, five relations as,

a) {(a, b) | a and b are the same age}

b) {(a, b) | a and b have the same parents}

c) {(a, b) | a and b share a common parent}

d) {(a, b) | a and b have met}

e) {(a, b) | a and b speak a common language}

we have to find which of them are equivalence relations

a) as, (a , a) ∈ R ∀ a hence, Reflexive.