Casio fx-991CW: The Inequalities Mode
Solving Inequalities
The Casio fx-991CW, along with other high-end scientific calculators from Casio has an Inequalities mode.
The inequalities mode solves quadratic, cubic, and quartic polynomials. The mode works similar to Equation-Polynomial module. Below is an example (all screenshots were created in Casio’s classpad.net website):
The Inequalities mode can assist in calculus and geometry problems.
Circle Problem
Task: Find valid points (x, y) that fit in the circle:
x^2 / 3 + y^2 / 6 = 1
Step 1: We can use the Inequalities mode to find the allowed x values.
Start by solving for y:
x^2 / 3 + y^2 / 6 = 1
6 × (x^2 / 3 + y^2 / 6) = 1 × 6
2 × x^2 + y^2 = 6
y^2 = 6 – 2 × x^2
y = √(6 – 2 × x^2)
For the equation to be “valid”, or to not return complex numbers, find all x where 6 – 2 × x^2 is not negative.
6 – 2 × x^2 ≥ 0
We find that the allow range is -√3 ≤ x ≤ √3.
Step 2: Store the Equations
We can do define f(x) in any mode. To do this, press [ f(x) ], select Define f(x). Press [ OK ] to store the function. The fx-991CW allows for two functions, f(x) and g(x).
In this example, we defined f(x) and g(x) as follows:
f(x) = √(6 – 2 × x^2)
g(x) = -f(x)
Step 3: Use the Table mode to find values in the appropriate range.
Set the Table Type as f(x)/g(x). Set the Table Range as Start: -√(3), End: √(3), Step: 2×√(3)÷5
Table:
|
x ≈ |
f(x) ≈ |
g(x) ≈ |
1 |
-1.732 |
0 |
0 |
2 |
-1.0392 |
1.9595 |
-1.9595 |
3 |
-0.3464 |
2.4 |
-2.4 |
4 |
0.3464 |
2.4 |
-2.4 |
5 |
1.0392 |
1.9595 |
-1.9595 |
6 |
1.732  |
0 |
0 |
The Area Between f(x) and x ≥ 0
Find the area between the curves:
f(x) = -x^4 + 2 × x^3 + 25 × x^2 – 26 × x – 120 and x ≥ 0.
Step 1: Find the boundaries.
With the Inequalities mode, we can find the boundaries and intersection points where f(x) ≥ 0. We find that: -4 ≤ x ≤ -2 and 3 ≤ x ≤ 5. Write down these results. The area will be in the intervals [-4, -2] and [3, 5].
Confession: I ultimately chose this function because of its integer roots. Unfortunately we are not able to store coefficients or roots in variables directly on the fx-991CW to variables.
Step 2: Store the Equations
Store -x^4 + 2 × x^3 + 25 × x^2 – 26 × x – 120 to f(x). This will allow us to save keystrokes while calculating the area.
Step 3: Calculate the Area in the Calculate Mode
Use the integral function from the Catalog - Func Analysis menu.
∫( f(x), -4, -2) + ∫( f(x), 3, 5)
The area is: 1928/15 ≈ 128.5333333
Hint: To always get a decimal approximation (skipping the exact result), press [SHIFT] [ = ] ( ≈ ).
Another example:
Find the area between the curves:
f(x) = x^3 + 8 × x^2 – 3 and x ≥ 0.
The boundaries are: -7.952564217 ≤ x ≤ -0.63837179, 0.5909359176 ≤ x
Which means the intervals are: [ -7.952564217, -0.63837179] and [ 0.5909359176, ∞). We can see that the area given these boundaries, the area will approach infinity.
Eddie
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