Fiend the dimensions of the rectangle garden producing the greatest enclosed area given 200 feet of fencing

For f(x)= cos²x + sinx, find the intervals where the function is increasing, decreasing, relative extrema, concavity, and points of inflection on the interval [0,2π)

(1 point) For the given position vectors r(t)r(t) compute the unit tangent vector T(t)T(t) for the given value of tt .

A) Let r(t)=〈cos5t,sin5t〉
Then T(π4)〈

B) Let r(t)=〈t^2,t^3〉
Then T(4)=〈

C) Let r(t)=e^(5t)i+e^(−4t)j+tk
Then T(−5)=

Optimal Chapter-Flight Fare

If exactly 180 people sign up for a charter flight, Leisure World Travel Agency charges $316/person. However, if more than 180 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency. Hint: Let x denote the number of passengers above 180. Show that the revenue function R is given by

R(x) = (180 + x)(316 − x).

passengers

What is the maximum revenue?

$

What would be the fare per passenger in this case?

dollars per passenger

Section 3.1 The Derivative and the Tangent Line Problem
Homework Assignment

   Find the derivative of the function by the limit process

   f(x)=1/x^2
   h(s)=-2√s

   Find the equation of the tangent line to the graph of the function at the given point.

   f(x)=x^3+1,(-1,0)
   f(x)=x-1/x,(1,0)

Section 3.2 Basic Differentiation Rules and Rates of Change
Homework Assignment
   Find the derivative of the function

   g(x)=∜x
   y=2x^3+6x^2-1
   g(t)=π cos⁡t
   y=7x^4+2 sin⁡x
   y=3/4 e^x+2 cos⁡x
   h(x)=(4x^3+2x+5)/x
   y=x^2 (2x^2-3x)
   y=3/〖(2x)〗^3 +2 sin⁡x
   g(x)=√x-3e^x
   Find the equation of the tangent line to the graph of the function f(x)=2/∜(x^3 ) at the point (1, 2).
Section 3.3 Product and Quotient Rules and Higher-Order Derivatives
Homework Assignment
   Find the derivative of the function

   g(s)=√s(s^2+8)
   g(x)=√x sin⁡x
   f(x)=x^2/(2√x+1)
   f(t)=cos⁡t/t^3
   y=sec⁡x/x
   f(x)=sin⁡x cos⁡x
   y=(2e^x)/(x^2+1)


   Find equation of the tangent line to the graph of the function f(x)=(x+3)/(x-3) at the point (4, 7)

a.)

Find the shortest distance from the point (0,1,2) to any point on the plane x - 2y +z = 2 by finding the function to optimize, finding its critical points and test for extreme values using the second derivative test.

b.)

Write the point on the plane whose distance to the point (0,1,2) is the shortest distance found in part a) above. All the work necessary to identify this point would be in part a). You just need to write the coordinates of the point here.

Find the exact location of all the relative and absolute extrema of the function. HINT [See Examples 1 and 3.] (Order your answers from smallest to largest x.) h(x) = (x − 1)2/3 with domain [0, 2] h has at (x, y) = . h has at (x, y) = . h has at (x, y) = .  

FOR ANSWER

please list the point and whether a

relative minimum a relative maximum an absolute minimum an absolute maximum no extremum at (t, y) =

A = [(α −β −β), (−β α −β), (−β −β α) ]

(each row is in paranthesis)

(c) Find the algebraic multiplicity and the geometric multiplicity of every eigenvalue of A.

(d) Justify if the matrix A is diagonalizable.

If $100,000 will purchase a 20-year annuity paying $739 at each month’s end, what monthly compounded nominal rate and effective rate of interest are earned by the funds? (Do not round intermediate calculations and round your final answers to 2 decimal places.)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

1)f(x, y) = 6y cos(x),    0 ≤ x ≤ 2π

2)f(x, y) = y² − 6y cos(x),    −1 ≤ x ≤ 7

3)f(x, y) = 6 sin(x) sin(y),     −π < x < π,     −π < y < π

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

1) f(x, y) = 2x³ − 6x + 6xy²

²⁾ f(x, y) = x³ + y³ − 3x² − 6y² − 9x

(1 point) A chain 66 meters long whose mass is 26 kilograms is hanging over the edge of a tall building and does not touch the ground. How much work is required to lift the top 8 meters of the chain to the top of the building? Use that the acceleration due to gravity is 9.8 meters per second squared. Your answer must include the correct units.

9.

a. If the unit of f(x) is gallons and the unit of x are in miles then the unit of f’(x) = ____________

b. If f is an increasing function at x = 4.5, then f ‘( 4.5) <0 True or False?

3. If g’ (4) = 0 and g “ (4) >0 , then g(4) is a ______________ value of the function.

4. If f(x) is continuous and f(2) = f(6) = 7 , then f’(c) =0 when 2<c<6. True or False?

5. If c(T) is the temperature (⁰F) of my coffee T minutes after I pour the cup, then explain in a sentence what c ‘(10) =-5 means? Don’t forget to include the unit.

Let f(x) = (x − 1)², g(x) = e^−2x, and h(x) = 1 + ln(1 − 2x).

L_f (x) = 

L_g(x) = 

L_h(x) = 

(b) Graph f, g, and h and their linear approximations. For which function is the linear approximation best? For which is it worst? Explain.

The linear approximation appears to be the best for the function  ? f g h since it is closer to  ? f g h for a larger domain than it is to  - Select - f and g g and h f and h . The approximation looks worst for  ? f g h since  ? f g h moves away from L faster than  - Select - f and g g and h f and h do.

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.04 cm thick to a hemispherical dome with diameter 44 m. (Round your answer to two decimal places.)
  m³