Graphical Sketch - Polynomials

Category: Analytic Geometry, Algebra

"Published in Suisun City, California, USA"

Sketch the graph for

Solution:

The first thing that we have to do is to factor the given equation by synthetic division as follows

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The x-intercepts are -1, 1, 2, and 3. To get the y-intercept, set x = 0 to the given equation, we have

Now, plot the points -1, 1, 2, and 3 along the x-axis and 6 along the y-axis. These points will pass the curve. To get the direction of the curve, let's consider the following

if    x < 3,       then y = (-)(-)(-)(-) = (+)
   - 3 < x < -1, then y = (-)(-)(+)(-) = (-)
    -1 < x < 1,  then y = (+)(-)(+)(-) = (+)
     1 < x < 2,  then y = (+)(-)(+)(+) = (-)
     x > 2,        then y = (+)(+)(+)(+) = (+)

Connect all the points and the graph must be like this

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Length - Four Leaf Rose

Category: Integral Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Suisun City, California, USA"

Find the length of a four leaf rose

Solution:

To draw the given equation in a polar coordinate system, we have to assign the values of θ from 0° to 360° in order to get their values of r as follows

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From the values of r and θ, we can sketch the graph of a four leaf rose as follows

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The length of a curve is given by the formula

Divide dx and dy by dθ as follows

Next, we need to eliminate dx and dy in the equation above as follows

If 

then

If

then

Therefore,

but 

The above equation becomes

The formula to get the length of any curve at polar coordinate system is

From the given problem that if

then

Since the given curve is symmetrical, we will use one quadrant only in getting the length of a curve and then multiply it later by 4. The range for the angle that we will use is from 0° to 90° or 0 to ½π as follows

Since the above equation is impossible to integrate by any method, then we have to use the Simpson's Rule in order to integrate the definite integral. Simpson's Rule is more accurate than Trapezoidal Rule and Riemann Sums. The formula for Simpson's Rule is

Let's consider the following, if n = 8, then

Next, let's tabulate the values of f(θ) as follows

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Therefore

The total length of a Four Leaf Rose is

Differentiation - Rate Problem

Category: Differential Calculus, Plane Geometry, Trigonometry

"Published in Newark, California, USA"

Each of two sides of a triangle are increasing at the rate of ½ foot per second, and the included angle is decreasing at 2° per second. Find the rate of change of the area when the sides and included angle are respectively 5 feet, 8 feet, and 60°.

Solution:

To illustrate the problem, it is better to draw the figure as follows

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From Plane Geometry, the area of a triangle is given by the formula

From Trigonometry, we know that

Therefore,

Take the derivative on both sides of the equation with respect to time, we have

but   b = 8 feet
        c = 5 feet
        θ = 60°
        ^db/_dt = ^dc/_dt = ½ ft/sec
        ^dθ/_dt = - 2°/sec (negative because decreases)
                = (- 2°/sec) x (π/180°) = - π/90 radians/sec

Substitute the values to the above equation, we have

Therefore,

Since the rate is positive, then the area is increasing.