A) Find the directional derivative of the function at the given point in the direction of vector v. f(x, y) = 5 + 6x√y, (5, 4), v = <8, -6>

Duf(5, 4) =

B) Find the directional derivative, D_uf, of the function at the given point in the direction of vector v.

f(x, y) =ln(x²+y²), (4, 5), v = <-5, 4>

D_uf(4, 5) =

C) Find the maximum rate of change of f at the given point and the direction in which it occurs.

f(x, y) =3 y²/x, (2, 4)

direction of maximum rate of change (in unit vector) = <    ,   >
maximum rate of change =

D) Find the directional derivative of f at the given point in the direction indicated by the angle θ.

f(x, y) = 2x sin(xy), (5, 0), θ = π/4

D_uf =

Calculate the second-order Taylor approximation to f(x, y) = 3 cos(x) sin(y) at the point (pi, pi/2)

1. What is a critical number of a function f ? What is the connection between critical numbers and relative extreme values?

2. What are inflection points? How do you find them?

Two parallel paths 20 m apart run east–west through the woods. Brooke walks east on one path at 7 km / h, while Jamail walks west on the other path at 5 km / h. If they pass each other at time t = 0 , how far apart are they 11 s later, and how fast is the distance between them changing at that moment?

The temperature at a point (x, y, z) is given by

T(x, y, z) = 100e^{−x2 − 3y2 − 7z2}

where T is measured in °C and x, y, z in meters.

(a) Find the rate of change of temperature at the point P(2, −1, 2) in the direction towards the point (4, −4, 4). answer in °C/m

(b) In which direction does the temperature increase fastest at P?

(c) Find the maximum rate of increase at P.

A deli sells 480 sandwiches per day at a price of $ 4 each. A market survey shows that for every $ 0.20 reduction in the original $ 4 ​price, 10 more sandwiches will be sold. Now how much should the deli charge in order to maximize​ revenue?

Please make sure it is the right answer. THanks!

Find the linearization at x=a.

f(x)=sin^7(x), a=π/4,

(Use symbolic notation and fractions where needed.)

Find the linearization of y=e^(√7x) at x=36.

(Use symbolic notation and fractions where needed.)

1. Find an equation of the circle that satisfies the given conditions.
Center (2, −3); radius 5

2. Find an equation of the circle that satisfies the given conditions.
Center at the origin; passes through (4, 6)

3. Find an equation of the circle that satisfies the given conditions.
Center (2, -10); tangent to the x-axis

4. Show that the equation represents a circle by rewriting it in standard form.
x² + y²+ 4x − 10y + 28 = 0

5. Show that the equation represents a circle by rewriting it in standard form.
x² + y² − 1/2 x + 1/2 y = 1/8

Illustrate Approaches that develop students’ imagery and visualization skills. (Illustrate means to draw or include pictures to represent.)

The carrying capacity for a country is 800 million. The population grew from 282 million in the year 2000 to 309 million in the year 2010.

1) Make a logistical model for determining the country's population

2) Use this model to predict the country's population in 2100

3) Predict the year in which the population will exceed 500 million

The curves r₁(t) = < t, 4, t²-9 > and r₂(s) = < 3, s², 4-2s >lie on surface of f and intersect at ( 3, 4, 0 ). Find a linear approximation for the # f(3.1, 4.1).

Find the volume V of the described solid S. The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Cross-sections perpendicular to the x−axis are squares.

V = ?

Consider the solid that lies above the rectangle (in the xy-plane) R=[−2,2]×[0,2],
and below the surface z=x²−4y+8.

(A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum.
Riemann sum =?

(B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the smallest possible Riemann sum.
Riemann sum =?

(C) Using iterated integrals, compute the exact value of the volume.
Volume = ?

A canned soup company wishes to design an optimal can for making a profit. The cost of the tin used for the can is 5 cents per cm^2. Suppose each can must contain 500 ml of soup. Find the dimensions of the can that minimize the cost. Justify your answer.