Problem 17 - The Calculus with Analytic Geometry

Question

\(y=3 x^{3}-5 x^{2}+1 ; x=t^{2}-1\)

Step-by-Step Solution

Answer

Substituting x = t^2 - 1 in the equation, the solution is y = 3t^6 - 14t^4 + 19t^2 - 7 .

Step 1 - Substitute the expression for x

Given that x = t^2 - 1 , substitute x in the main function y = 3x^3 - 5x^2 + 1 .

Step 2 - Expand the expression

Expand the substituted expression by replacing x with t^2 - 1 in y = 3(t^2 - 1)^3 - 5(t^2 - 1)^2 + 1 .

Step 3 - Simplify (t^2 - 1)^3

Simplify (t^2 - 1)^3to get t^6 - 3t^4 + 3t^2 - 1 .

Step 4 - Simplify (t^2 - 1)^2

Simplify (t^2 - 1)^2to get t^4 - 2t^2 + 1 .

Step 5 - Substitute back and combine like terms

Now, substitute the simplified expressions back into the main equation: y = 3(t^6 - 3t^4 + 3t^2 - 1) - 5(t^4 - 2t^2 + 1) + 1 . Combine the like terms to simplify completely.

Step 6 - Final Simplification

Combine and simplify: y = 3t^6 - 9t^4 + 9t^2 - 3 - 5t^4 + 10t^2 - 5 + 1 . Therefore, y = 3t^6 - 14t^4 + 19t^2 - 7 .

Key Concepts

substitution method

The substitution method is a useful technique in mathematics, allowing us to replace one variable with another to simplify the problem at hand.
In this exercise, we are given two equations: one for y and one for x. The equation for x is x = t^2 - 1. To simplify the equation for y, we substitute this expression into y's equation:
y = 3x^3 - 5x^2 + 1.
Let's look at how this substitution transforms the equation:

• Original equation: y = 3x^3 - 5x^2 + 1
• Substitute x: y = 3(t^2 - 1)^3 - 5(t^2 - 1)^2 + 1

By replacing x with t^2 - 1, we have moved from a problem in terms of x to one in terms of t. This opens the door to other methods for simplifying our work, such as polynomial expansion and combining like terms.

polynomial expansion

Polynomial expansion involves expressing a polynomial raised to a power in a more detailed, expanded form.
It's crucial to manage the expressions carefully to simplify effectively. In our substituted equation, we need to expand two expressions:

• (t^2 - 1)^3
• (t^2 - 1)^2

Let’s start by expanding (t^2 - 1)^3:
(t^2 - 1)^3 is expanded as: (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3, giving us:
Now, let's expand (t^2 - 1)^2
(t^2 - 1)^2 is expanded as: (a - b)^2 = a^2 - 2ab + b^2, giving us:
With both polynomial expansions completed, we return to our expression for y: y = 3(t^6 - 3t^4 + 3t^2 - 1) - 5(t^4 - 2t^2 + 1) + 1
Our next step will be to simplify this expression.

simplification

Simplification is the process of condensing an expression to its simplest form.
After expanding the polynomial, we gather all terms to simplify:
Starting with: y = 3(t^6 - 3t^4 + 3t^2 - 1) - 5(t^4 - 2t^2 + 1) + 1

• Distribute 3 into the first set of terms:
• y = 3t^6 - 9t^4 + 9t^2 - 3


• Distribute -5 into the second set of terms:
• y = -5t^4 + 10t^2 - 5

Collect all terms and constants:
• y = 3t^6 - 9t^4 + 9t^2 - 3 - 5t^4 + 10t^2 - 5 + 1

Combine like terms:
y = 3t^6 - 14t^4 + 19t^2 - 7
Thus, the simplified equation for y in terms of t is: y = 3t^6 - 14t^4 + 19t^2 - 7.
This process highlights the power of combining substitution, polynomial expansion, and simplification to reduce complex equations into something more manageable.