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Magazine Chapter 23: Problem 35
Solve the given problems by finding the appropriate derivatives. Using the product rule, find the point(s) on the curve of \(y=\left(2 x^{2}-1\right)(1-4 x)\) for which the tangent line is \(y=4 x-1\).
Short Answer
Expert verified
The points are \((0, -1)\) and \(\left(\frac{1}{6}, -\frac{19}{18}\right)\)."
Step by step solution
01
Define the Function and Derivative
We are given the function \( y = (2x^2 - 1)(1 - 4x) \). To find the points where the tangent line equals \( y = 4x - 1 \), we need to find the derivative of this function using the product rule. The product rule states \((uv)' = u'v + uv'\), where \( u = 2x^2 - 1 \) and \( v = 1 - 4x \).
02
Find the Derivatives of u and v
Differentiate \( u = 2x^2 - 1 \) with respect to \( x \) to get \( u' = 4x \). Then, differentiate \( v = 1 - 4x \) with respect to \( x \) to get \( v' = -4 \).
03
Apply the Product Rule
Substitute \( u \), \( u' \), \( v \), and \( v' \) into the product rule formula: \[ \frac{dy}{dx} = (4x)(1 - 4x) + (2x^2 - 1)(-4) \]. Simplify the expression to find the derivative of the function.
04
Simplify the Derivative
Simplify \( \frac{dy}{dx} = 4x - 16x^2 - 8x^2 + 4 \). Combine like terms to get \( \frac{dy}{dx} = -24x^2 + 4x + 4 \).
05
Set Derivative Equal to Tangent Slope
Set the derivative \( \frac{dy}{dx} = -24x^2 + 4x + 4 \) equal to the slope of the tangent line given in the problem, which is \( 4 \). Solve for \( x \) in the equation \( -24x^2 + 4x + 4 = 4 \).
06
Solve the Equation
Subtract \( 4 \) from both sides: \( -24x^2 + 4x = 0 \). Factor the equation: \( 4x(-6x + 1) = 0 \). This gives two solutions: \( x = 0 \) and \( -6x + 1 = 0 \) leading to \( x = \frac{1}{6} \).
07
Find the Corresponding y-values
For \( x = 0 \), substitute back into the original function to get \( y = (2(0)^2 - 1)(1 - 4(0)) = -1 \). For \( x = \frac{1}{6} \), substitute back to get \( y = \left(2\left(\frac{1}{6}\right)^2 - 1\right)(1 - 4\left(\frac{1}{6}\right)) = -\frac{19}{18} \). Thus, the points are \((0, -1)\) and \(\left(\frac{1}{6}, -\frac{19}{18}\right)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Product Rule
When dealing with a function that is a product of two other functions, it's time to use the product rule. This rule is a handy tool in calculus for finding the derivative of such functions. The product rule formula \( (uv)' = u'v + uv' \) is used to differentiate the product of two functions \( u(x) \) and \( v(x) \).
Let's break it down:
Let's break it down:
- \( u \) and \( v \) are the two functions being multiplied.
- \( u' \) is the derivative of \( u \).
- \( v' \) is the derivative of \( v \).
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Think of it as a line that just skims the surface of the curve.
- The tangent line's slope at a particular point is the same as the slope of the curve at that point.
- In derivative terms, this slope is represented by the derivative of the function at that point.
Differentiation
Differentiation is a core concept in calculus, used to determine how a function changes as its input changes. It's essentially the process of finding a derivative. Derivatives tell us the slope of a function at any point.
There are key steps in differentiation:
There are key steps in differentiation:
- Identify the function you want to differentiate.
- Use rules like the product rule, quotient rule, or chain rule depending on the nature of the function.
- Simplify the expression to arrive at a clean derivative.
Slope of a Curve
The slope of a curve at a particular point describes how steep the curve is at that very point. Unlike a straight line that has constant slope, a curve changes slope continuously.
The slope of the curve at a point is given by the derivative of the function at that point. The steeper the curve, the larger the magnitude of the slope; similarly, flatter curves have slopes closer to zero.
The slope of the curve at a point is given by the derivative of the function at that point. The steeper the curve, the larger the magnitude of the slope; similarly, flatter curves have slopes closer to zero.
- Positive slope: means the function is increasing.
- Negative slope: means the function is decreasing.
- Zero slope: indicates a flat point, which might be a local maximum or minimum.
