Estimate the area under the graph of

f(x)=25−x^2

from x=0 to x=5 using 5 approximating rectangles and right endpoints.

(B) Repeat part (A) using left endpoints.
(C) Repeat part (A) using midpoints.

Evaluate each integral using trig substitutions

1.) Integral of (3x^5dx)/(sqrt(16-x^2)

2.) Integral of (sqrt(x^2-16)dx)/x

3.) Integral of (6dx)/(16+16x^2)

1.The position of a particle in rectilinear motion is given by s (t) = 3sen (t) + t ^ 2 + 7. Find the speed of the particle when its acceleration is zero.

2.Approximate the area bounded by the graph of y = -x ^ 2 + x + 2, the y-axis, the x-axis, and the line x = 2.
a) Using Reimmann sums with 4 subintervals and the extreme points on the right.
b) Using Reimmann sums with 4 subintervals and the extreme points on the left.

Submit the sum of parts a) and b) in response.

5. Solve equation y'+2y=2-e^-4t, where y(0)=1

6. Use Euler’s method for a previous problem at t=0, 0.1, 0.2, 0.3. Compare approximate and the exact values of y.

Define vector addition as y_1⊕ y_2=y_1+y_2-3 and scalar multiplication as c ⊙y_1=cy_1-3c+3. Let V be the space of all solutions of the equation in problem 3 that uses the above definitions of vector addition and scalar multiplication. Use the Vector Space Definition to prove that V is a vector space

Does every polynomial equation have at least one real root?

a. Why must every polynomial equation of degree 3 have at least one real root?

b. Provide an example of a polynomial of degree 3 with three real roots. How did you find this?

c. Provide an example of a polynomial of degree 3 with only one real root. How did you find this?

Let g(t) = 5t − 3 ln(t 2 ) − 4(t − 2)2 , 0.1 ≤ t ≤ 3.

a. Find the absolute maximum and minimum of g(t).

b. On what intervals is g(t) concave up? Concave down?

Given two lines in​ space, either they are​ parallel, they​ intersect, or they are skew​ (lie in parallel​ planes). Determine whether the lines​ below, taken two at a​ time, are parallel, intersect, or are skew. If they​ intersect, find the point of intersection.​ Otherwise, find the distance between the two lines.

L1: x=1-t, y=-2-2t, z=1-t

L2: x=-1+2s, y=-5+4s, z=1+2s

L3: x=-1+2r, y=-5+3r, z=2-r

Find for L1 and L2, L2 and L3, L1 and L3.

Have you ever hear about Banach Spaces? Chances are, you never had. Yet, they are an essential part of today’s mathematics. Please search some info about the person that first introduced them to mathematics, Stefan Banach.

P= 16x -5y +66, subject to 7x +9y ≤ 63, 0 ≤ y ≤ 4, and 0 ≤ x ≤ 5.

The maxium value _____ occurs when x= _____ and y= ______.

The minimun value ____ occurs where x= ____ and y=____.

The figure shows four circles, each with a radius of 6 cm. Find the area of the region between the circles. (Round your answer to two decimal places.)
cm²

A student cuts out a circle from a square piece of cardboard. The circle passes through the midpoints of the sides of a square as shown. Each side of the square has a length of 12 units. What percent of the square cardboard is wasted? (Round your answer to two decimal places.)

36.34%27.32%    21.46%25.00%28.54%

A small bucket of paint covers 145 square feet. How many buckets of paint would you need to paint a rectangular wall that is 44 by 17 feet?

7 buckets of paint1 bucket of paint    748 buckets of paint5 buckets of paint6 buckets of paint

A regular hexagon has an apothem of 8.7 cm and perimeter of 60 cm.

(a) What is the area of the hexagon? (Round your answer to the nearest square centimeter.)

A = 87 cm²A = 261 cm²     A = 1044 cm²A = 904 cm²A = 522 cm²

(b) What is the radius of the hexagon? (Round your answer to the nearest hundredth of a centimeter.)

Use a formula to find the area of the triangle.
square units

r = 10.03 cmr = 7.12 cm    r = 4.93 cmr = 13.25 cmr = 13.61 cm

Find the 5th Taylor polynomial of f(x) = 1+x+2x^5 +sin(x^2) based at b = 0.

Draw the graph, solid of revolution, one representative disk/ washer.

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region formed by

a) when revolved about y-axis, the volume is ?

b) when revolved about x-axis, the volume is ?

c) when revolved about the line y=8, the volume is ?

d) when revolved about the line x=2, the volume is ?

a. Verify that the given point lies on the curve.

b. Determine an equation of the line tangent to the curve at the given point.

9 (x² y²)² =100xy² ; (1,3)

a)

Let y be the solution of the equation  y ′ = (y/x)+1+(y^2/x^2) satisfying the condition  y ( 1 ) = 0. Find the value of the function  f ( x ) = (y ( x ))/x

at  x = e^(pi/4) .

b)

Let y be the solution of the equation  y ′ = (y/x) − (y^2/x^2) satisfying the condition  y ( 1 ) = 1. Find the value of the function  f ( x ) = x/(y(x))

at  x = e  .

c)

Let y be the solution of the equation

y ′ + (3x^2*y)/(1+x^3)=e^x/(1+x^3) satisfying the condition  y ( 0 ) = 1.

Find  ln ⁡ ( 2 y ( 1 ) ).