Problem 18 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find \(c\) so that the curve \(y=4-x-c x^{2}\) passes through the point \((-2,1)\).

Step-by-Step Solution

Answer

The value of \(c\) is \(\frac{5}{4}\).

Step 1 - Substitute the given point into the equation

The point \((-2, 1)\) must satisfy the equation \(y = 4 - x - c x^2\). Substitute \(-2\) for \(x\) and \(1\) for \(y\) in the equation: \(1 = 4 - (-2) - c (-2)^2\).

Step 2 - Simplify the equation

Simplify the substituted equation. This results in \(1 = 4 + 2 - 4c\).

Step 3 - Solve for \(c\)

Combine the constants and solve for \(c\): \(1 = 6 - 4c \Rightarrow 4c = 5 \Rightarrow c = \frac{5}{4}\).

Key Concepts

Substitution Method

The substitution method is a crucial concept in calculus and algebra. It involves substituting known values into an equation to find an unknown variable.
The first step in solving this problem is to substitute the given point \((-2, 1)\) into the curve's equation. This is because the point must satisfy the equation. By doing this:

• Replace \(x\) with -2
• Replace \(y\) with 1

This step transforms the original equation \(y = 4 - x - c x^2\) into \(1 = 4 - (-2) - c (-2)^2\). Substitution helps link the given point to the parameters of the curve, paving the way for further simplification.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system. It allows us to solve problems involving curves and points through algebraic methods.
In this exercise, we apply coordinate geometry by ensuring that the point \((-2, 1)\) lies on the curve. This concept is essential because:

• It helps us understand the relationship between algebraic equations and geometric figures, such as parabolas or lines.
• By applying the point to the equation, we can derive mathematical expressions to describe the curve's properties.

Understanding the principles of coordinate geometry simplifies the problem, allowing us to find the correct value of \(c\).

Equation Simplification

Simplifying equations is a vital math skill. It involves combining like terms and reducing complex expressions to simpler forms.
In this problem, once we've substituted the point into the equation, we need to simplify the resulting expression \1 = 4 + 2 - 4c\. The simplifying process includes:

• Adding up the constants to get \1 = 6 - 4c
• Isolating the variable \(c\) by rearranging and solving the simplified equation

The simplification ultimately leads us to solve \4c = 5\, giving the solution \c = \frac{5}{4}\.
This step-by-step process ensures clarity and aids in understanding how each part of the equation affects the final result.