The three parts of this problem concern the area under the graph of y=f(x)=1/x on the interval (.5,4). (A) Estimate this area using L6. That is, use 6 equal width rectangles over the interval (.5,4), the height of each rectangle determined by the left endpoint of each subinterval. (B) Estimate this area using R6. That is, use 6 equal width rectangles over the interval (.5,4) with the height of each rectangle determined by the right endpoint of each subinterval. (C) Use calculus methods to determine the exact area by evaluating the definite integral (.5,4) 1/x dx

1) Find the largest value of x that satisfies:
log5(x2)−log5(x+5)=8

2) Students in a fifth-grade class were given an exam. During the next 2 years, the same students were retested several times. The average score was given by the model
f(t)=90−7log(t+1),    0≤t≤24
where t is the time in months. Round answers to at least 1 decimal point.

A) What is the average score on the original exam?

B) What was the average score after 6 months?

C) What was the average score after 18 months?

1. Provide the equation of the line that has slope ? = −4 and passes through the point (−2,1).

2.Rewrite ?(?) = |? − 4| as a piecewise-defined function.

3.Rewrite ln((?4(?2+2) )/??3 ) in terms of simpler logarithms (no powers, products, or quotients inside the logarithm).

4.Test ℎ(?) = −2?^4 + 8? − 3 for symmetry and state your conclusion.

5. If the domain is restricted to the open interval (-pi/2,pi/2),find the range of f(x)=e^tan x

In a football game, Team A defeated Team B by a score of 51 to 44. The total points scored came from 28 different scoring plays, which were a combination of touchdowns, extra-point kicks, field goals, and safeties, worth 6, 1, 3, and 2 points, respectively. There were four times as many touchdowns as field goals, and the number of extra-point kicks was equal to the number of touchdowns. How many touchdowns, extra-point kicks, field goals, and safeties were scored during the game?

Given: Polynomial P(x) of degree 6

Given: x=3 is a zero for the Polynomial above

List all combinations of real and complex zeros, but do not consider multiplicity for the zeros.

Find the curvature of curve r(t) = 7sin2ti+7cos2tj+7tk

5. Use the Laws of Logarithms to combine the expression: 3 (log5 (x) + 3 log 5 (y)−4 log 5 (z) )​

The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 2 m and w = h = 3 m, and l and w are increasing at a rate of 7 m/s while h is decreasing at a rate of 5 m/s. At that instant find the rates at which the following quantities are changing.

(a) The volume.
m³/s

(b) The surface area.
m²/s

(c) The length of a diagonal. (Round the answer to two decimal places.)
m/s

The monthly sales of Stewart Electronics' new sound system are given by q(t) = 2000t - 100t² units per month, t months after its introduction. The price Stewart charges is p(t) = 1000 - t² dollars per sound system, t months after introduction.

a. Find the rate of change of monthly sales after 6 months.

b. Find the rate of change of monthly price after 6 months.

c. Find the equation of the rate of change of the monthly revenue.

d. Find the rate of change of the monthly revenue after 6 months.

1. Using Table 1---Find the Future Value of $5,000 at 6% interest compounded semi-annually for 2 years.

$5,627.55

$5.835.64

$5,527.55

$5,735.64

2.Using Table 2---Find the Compound Interest on $2,500 at .75% interest compounded daily by Leader Financial Bank for 20 days.

$1.02

$10.30

$1.03

$6.31

A box with an open top is to be constructed out of a rectangular piece of cardboard with dimensions length=10 ft and width=11 ft by cutting a square piece out of each corner and turning the sides up. Determine the length x of each side of the square that should be cut which would maximize the volume of the box.

An open-top rectangular box is to be constructed with 300 in2 of material. If the bottom of the box forms a square, what is the largest possible box, in terms of volume, that can be constructed?

1. You plan to calculate the volume of a tall, thin cylinder from measurements of its radius and height. Which dimension should you measure more carefully? Give reasons for your answer.