# Define Bounded Area Formula

Define A(x) to be the area bounded by the t-axis, the line y = 2t and a vertical line at t = x.

(a) Find a formula for A(x).

(b) Determine A'(x)

The figure below shows the graph of the derivative of a continuous function f .

(a) List the critical numbers of f .

(b) What values of x result in a local maximum?

(c) What values of x result in a local minimum?

Use information from the derivative of each function to help you graph the function. Find all local maximums and minimums of each function.

g(x)=2x^3-〖15x〗^2+6

In 4 and 5, a function and values of x so that f '(x) = 0 are given. Use the Second Derivative Test to determine whether each point (x, f (x)) is a local maximum, a local minimum or neither.

h(x)=x^4-8x^2-2; x=-2,0,2

f(x)=x∙In(x); x=1/e

Lest you have forgotten, the formulas you will need are as follows:

For a right circular cylinder of radius "r" and height "h":

The volume V = πr2h.

The surface area S = circular ends plus the cylindrical wall = 2πr2 + 2πrh.

You have been asked to design a one-liter (i.e., 1000 cm3) can shaped like a right circular cylinder (figure below). What dimensions will use the least material?

So the question is: What should "r" be and what should "h" be such that the volume is 1 liter but the surface area is least?

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#### Solution Preview

1. Define A(x) to be the area bounded by the t-axis, the line y = 2t and a vertical line at t = x.

(a) Find a formula for A(x).

(b) Determine (x)

Solution:

(a)

(b)

2. The figure below shows the graph of the derivative of a continuous function f .

(a) List the critical numbers of f .

(b) What values of x result in a local maximum?

(c) What values of x result in a local minimum?

Solution:

(a)

Critical numbers are x = 2, 7

(b)

Function has local maximum at x = 7.

(c)

Function has local minimum at x = 2.

3. Use information from the derivative of each function to help you graph the function. Find all local maximums and minimums of each function.

Solution:

To find critical points, put g'(x) = 0

Critical points are x = 0, 5

Since g"(0) < 0 ...

#### Solution Summary

This posting provides step by step instructions to some questions on finding local maximum/minimum.

derivative and the slope of a curve

In this explanation, I am being asked to discuss the relationship between the slope of a secant line, the slope of a tangent line and the derivative AND in addition, I must explain the relationship between the area of a finite number of rectangles under a curve and an infinite number of rectangles under a curve and the definite integral.

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