Find an approximation of the area of the region R under the graph of the function f on the interval [1, 3]. Use n = 4 subintervals. Choose the representative points to be the right endpoints of the subintervals.

f(x)=6/x

Find the area of the region under the graph of the function f on the interval [3, 7].

Find the area of the region under the graph of the function f on the interval [-27, -1].

Find the area of the region under the graph of f on  

[a, b].

f(x) = 5/x^2;    [1, 2]

Find the area of the region under the graph of the function f on the interval [1, 8].

f(x)=10/x

Find the area of the region under the graph of the function f on the interval [0, 5].

f(x) = 5x -x^2

ALL OF THESE ARE PART OF THE SAME PROBLEM

1. rewrite as a single logarithm: 4ln(x-2x)-3lnx+ln7

2. Solve using common base: 4^3x-1=16⁵

^{3. Solve using logarithm:} 9^4x-3=845

4. 6log₄(7x)=-18

1. Use the derivative function, f'(x)f′(x), to determine where the function

f(x)=−2x^2+14x−8

is increasing.

2.Use the derivative function f'(x)f′(x) to determine where the function f(x)=2x^3−27x^2+108x+13 is increasing.  

3.Use the derivative function f'(x)f′(x) to determine where the function f(x)=2x^3−27x^2+108x−12 is decreasing.

4.Find each value of the function f(x)=−x^3+12x+9 where the line tangent to the graph is horizontal.

x=

Let f(x, y) = − cos(x + y² ) and let a be the point a = ( π/2, 0).

(a) Find the direction in which f increases most quickly at the point a.

(b) Find the directional derivative Duf(a) of f at a in the direction u = (−5/13 , 12/13) .

(c) Use Taylor’s formula to calculate a quadratic approximation to f at a.

. Let two circles C1 and C2 intersect at X and Y. Prove that a point P is on the line XY if and only if the power of P with respect to C1 is equal to the power of P with respect to C

1. Which of the following is the linear approximation of the function f ( x ) = 2e^sin (7x) at x = 0?

Group of answer choices

y=cos⁡(7)x+2

y=7x+2

y=14x+2

y=2x+7

y=e^7x+14

2. Recall that Rolle's Theorem begins, ``If f ( x ) is continuous on an interval [ a , b ] and differentiable on (a , b) and ___________, then there exists a number c …'' Find all values x = c that satisfy the conclusion of Rolle's Theorem for the function h ( x ) = x^3 − 7x − 9 on the interval [ − 2 , − 1 ].

3. At x = 1, the function g( x ) = 5x ln(x) − 3/x

is . . .

Group of answer choices

has a critical point and is concave up

decreasing and concave up

decreasing and concave down

increasing and concave up

increasing and concave down

4.If x and y are two real numbers such that 8x − y = 48, what is the smallest possible value of their product xy?

Group of answer choices

12

−72

0

−24

−144

Matlab code please

6. Find the velocity, acceleration, and speed of a particle with the given position function. (a) r(t) = e t cos(t)i+e t sin(t)j+ tetk t = 0 (b) r(t) = 〈t 5 ,sin(t)+ t 2 cos(t),cos(t)+ t 2 sin(t)〉, t = 1

1. Find the intervals on which the graph of f is concave​ upward, the intervals on which the graph of f is concave​ downward, and the inflection points. f(x)= -x^4 + 12x^3 - 12x + 19 For what​ interval(s) of x is the graph of f concave​ upward?

2. For the function ​f(x)= (8x-7)^5

a. The​ interval(s) for which​ f(x) is concave up.

b. The​ interval(s) for which​ f(x) is concave down.

c. The​ point(s) of inflection.

Derivatives of Exponential Functions (Scenario)

Consider the following problem: Suppose that F(x) computes the rabbit population on a game reserve that doubles every 6 months. Suppose there were 200 rabbits initially.

1. Write a mathematical expression for F(x).

2. Find domain and range of F(x) and discuss its meaning

3. Find F(x) and F'(x) at any point and discuss its physical meaning

4. Find all x values for which F'(x)=0 and discuss what this function means

5. Discuss if your function F(x) is differentiable and why. If it is not, select another function that is and discuss the change you made.

6. Discuss the criteria for selecting a real-world scenario that would change if you were seeking to model it with a logarithmic function instead. What key similarities and differences would you find?

***Please discuss why...I am having a hard time understanding each meaning with these scenarios*** Thank You!!!

If f(x)=2x^2−5x+3, find f'(−4).     

Use this to find the equation of the tangent line to the parabola y=2x^2−5x+3 at the point (−4,55). The equation of this tangent line can be written in the form y=mx+b
where m is: ????

and where b is: ????

Evaluate the line integral ∫CF⋅dr∫CF⋅dr, where F(x,y,z)=sin(x)i+cos(y)j+4xzkF(x,y,z)=sin(x)i+cos(y)j+4xzk and C is given by the vector function r(t)=t3i−t2j+tkr(t)=t3i−t2j+tk , 0≤t≤10≤t≤1.

Find the absolute extrema if they exists, as well as all values of x, where they occur, for the function f(x)=1/3x^3+1/2x^2-12x+5 on domain [6,6]

The following table provides the number of men infected with HIV in San Francisco from 1982 to 1990. Year Quantity Delaware infected 1982 80 1983 300 1984 700 1985 1500 1986 2500 1987 3,500 1988 4500 1989 6000 1990 7200 Estimate the inflection point of the data and try a brief explanation of its meaning in this context.

1. A spherical balloon is being inflated so that its volume increases at a rate of 1200cm^3 per minute. How fast is its radius increasing at the moment when the radius is 20cm? Recall that the volume of a sphere is given by V = 4/3πr^3

2. Which of the following is the linear approximation of the function f ( x ) = 2e^sin (7x) at x = 0?

Group of answer choices

y=cos⁡(7)x+2

y=7x+2

y=14x+2

y=2x+7

y=e^7x+14

3. Recall that Rolle's Theorem begins, ``If f ( x ) is continuous on an interval [ a , b ] and differentiable on ( a , b ) and ___________, then there exists a number c …'' Find all values x = c that satisfy the conclusion of Rolle's Theorem for the function h ( x ) = x^3 − 7x − 9 on the interval [ − 2 , − 1 ].

4.If x and y are two real numbers such that 8x − y = 48, what is the smallest possible value of their product xy?

Group of answer choices

12

−72

0

−24

−144

Find the value of Δz for the function f(x,y)=2x2+xy2+y at point (1,1). Use Δx=0.02 and Δy=0.02. Round to three decimal places.