Let f(x,y)=x^2 + ln(y).Calculate the instantaneous rate of change at (3,1) to (1,2).

Answers

Answer 1

Answer:

To find the instantaneous rate of change of the function f(x,y) = x^2 + ln(y) at (3,1) to (1,2), we can use the partial derivatives with respect to x and y:

fx(x,y) = 2x

fy(x,y) = 1/y

Then, we can use the gradient vector to find the direction of maximum increase:

∇f(x,y) = <fx(x,y), fy(x,y)> = <2x, 1/y>

At point (3,1), the gradient vector is:

∇f(3,1) = <6, 1>

At point (1,2), the gradient vector is:

∇f(1,2) = <2, 1/2>

To find the instantaneous rate of change from (3,1) to (1,2), we can use the formula for directional derivative:

Dv(f) = ∇f(x,y) · v

where v is the unit vector in the direction from (3,1) to (1,2). The direction vector v is given by:

v = <1, 2> - <3, 1> = <-2, 1>

To make v a unit vector, we need to normalize it by dividing it by its length:

|v| = sqrt((-2)^2 + 1^2) = sqrt(5)

u = v/|v| = <-2/sqrt(5), 1/sqrt(5)>

Then, the instantaneous rate of change from (3,1) to (1,2) is:

Dv(f) = ∇f(3,1) · u = <6, 1> · <-2/sqrt(5), 1/sqrt(5)> = (-12/sqrt(5)) + (1/sqrt(5)) = -11/sqrt(5)

Therefore, the instantaneous rate of change of the function f(x,y) = x^2 + ln(y) from (3,1) to (1,2) is -11/sqrt(5).

To learn more about instantaneous rate of change refer below:

brainly.com/question/31011769

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(Will give Brainliest) Please HelppA pizza parlor makes 52 pizza crusts the first week of summer and 180 pizza crusts each subsequent week. A diner makes 26 pizza crusts the first week of summer and 90 pizza crusts each subsequent week. In how many weeks will the total number of pizza crusts made by the pizza parlor be twice the total number of pizza crusts made by the diner?

The total number of pizza crusts made by the pizza parlor will_______

Answers

Answer:

Every week

Step-by-step explanation:

Let the number of weeks be x, then equations

Pizza parlor

52 + 180x

Diner

26 +90x

In how many weeks will the total number of pizza crusts made by the pizza parlor be twice the total number of pizza crusts made by the diner?

52 + 180x = 2(26 +90x)
52 + 180x = 52 + 180x
0 = 0

This results means that every week pizza parlor makes twice the number of by diner

Answer:

$\large \boxed{\sf always \ be \ double \ than \ the \ diner}$

Step-by-step explanation:

Let x represent the no. of weeks

Pizza parlor: 180x + 52

Diner: 90x + 26

Write the equation and solve for x

180x + 52 = 2(90x + 26)

x = x (TRUE)

The total number of pizza crusts made by the pizza parlor will always be double than the diner.

What is base 1 of a trapezoid in which A=(48x+68) in^2, height =8, and base 2 = (9x + 12)

Answers

A=(1/2)(b1+b2)h =

=(48x+68)in² = (1/2)( b1+(9x+12))8

the answer is = b1= 3x+5

i hope that helped =D

the figure below shows segments ac and ef which intersect at point b. segment af is parallel to segment ec: which of these facts is used to prove that triangle abf is similar to triangle cbe? angle fab is equal to angle ceb because corresponding angles are congruent. angle abf is congruent to angle ceb because vertically opposite angles are congruent. angle afb is congruent to angle ceb because alternate interior angles are congruent. angle afb is congruent to angle ceb because supplementary angles are congruent. i chose b, but i'm not positive it is the answer. can anyone give some clarification?