Problem 133 - Calculus Volume 1

Question

For the following exercises, a. evaluate \(f^{\prime}(a),\) and b. graph the function \(f(x)\) and the tangent line at \(x=a\) $$\mathrm{[T]} f(x)=2 x^{3}+3 x-x^{2}, a=2$$

Step-by-Step Solution

Answer

a) \(f'(2) = 23\); b) Graph \(f(x)\) and tangent line \(y=23x-28\).

Step 1: Find the derivative of f(x)

To find the derivative \( f'(x) \), apply the power rule to each term of \( f(x) = 2x^3 + 3x - x^2 \). The derivative of \( x^n \) is \( nx^{n-1} \). Thus, \( f'(x) = 6x^2 + 3 - 2x \).

Step 2: Evaluate f'(a)

Now that we have \( f'(x) = 6x^2 + 3 - 2x \), substitute \( a = 2 \) to find \( f'(2) \). Calculate: \( f'(2) = 6(2)^2 + 3 - 2(2) = 24 + 3 - 4 = 23 \). Therefore, \( f'(2) = 23 \).

Step 3: Find the point on the curve

Substitute \( x = 2 \) into the original function \( f(x) = 2x^3 + 3x - x^2 \) to find the y-coordinate. \( f(2) = 2(2)^3 + 3(2) - (2)^2 = 16 + 6 - 4 = 18 \). The point on the curve is \( (2, 18) \).

Step 4: Write the equation of the tangent line

Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( m = f'(a) = 23 \) and \( (x_1, y_1) = (2, 18) \), write the equation of the tangent line: \( y - 18 = 23(x - 2) \). Simplify it to \( y = 23x - 46 + 18 \), which gives \( y = 23x - 28 \).

Step 5: Graph f(x) and the tangent line

Plot \( f(x) = 2x^3 + 3x - x^2 \) by evaluating a few key points and noting the cubic shape. Plot the tangent line \( y = 23x - 28 \) which should just touch the curve at \( x = 2 \). Verify that the tangent line intersects the curve at the point \( (2, 18) \).

Key Concepts

Understanding the Power Rule

In calculus, derivatives are essential for finding rates of change. A popular method to find derivatives is the power rule. The power rule helps us quickly determine the derivative of a function that is expressed as a power of a variable.
For a function of the form \( f(x) = ax^n \), the derivative \( f'(x) \) is found by multiplying the exponent \( n \) by the coefficient \( a \), and then reducing the exponent by one.
In mathematical terms, the derivative is \( f'(x) = nax^{n-1} \). This rule is very simple and quick to apply. For example, in our function \( f(x) = 2x^3 + 3x - x^2 \), the power rule allows us to find \( f'(x) = 6x^2 + 3 - 2x \) by:

• Calculating the derivative of \( 2x^3 \) which is \( 6x^2 \).
• The derivative of \( x^2 \) results in \( 2x \).

So, diminishing complex calculations became straightforward using the power rule.

What is a Tangent Line?

A tangent line is a straight line that touches a curve at only one point. At this point, the tangent line has the same slope as the curve.
This means that the tangent line gives us a linear approximation of the curve at the point of tangency, which can be very useful.
To find the equation of a tangent line, you need a specific point on the curve and the slope of the curve at that point.
When analyzing the function \( f(x) = 2x^3 + 3x - x^2 \), the tangent line at \( x=2 \) needs:

• The point on the curve, which is \( (2, 18) \).
• The slope of the curve at \( x=2 \), known as \( f'(2) = 23 \).

The tangent line equation then becomes \( y - 18 = 23(x - 2) \), simplified to \( y = 23x - 28 \), perfectly mirroring the slope of \( f(x) \) at \( x=2 \).

Exploring Cubic Functions

Cubic functions are polynomial functions of degree three. They have the general form \( f(x) = ax^3 + bx^2 + cx + d \).
Such functions are known for their 'S' shaped curves and the possibility of having multiple turning points and points of inflection.
Analyzing the cubic function \( f(x) = 2x^3 + 3x - x^2 \) lets us explore these features more closely.
Cubic functions can have:

• Zero to three real roots—the points where the function intersects the x-axis.
• One or two turning points—where the function changes direction from increasing to decreasing or vice versa.

In our exercise, the turning point analysis helped us understand where the tangent line \( y = 23x - 28 \) fits, especially since the function's behavior is reflective of its derivative at \( x=2 \). This makes cubic functions quite dynamic and exciting for deeper studies.

Using the Point-Slope Form Effectively

The point-slope form is a convenient way to write the equation of a line when you have a specific point and a slope. This form is given by \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \( (x_1, y_1) \) is a point the line passes through.
This form is particularly useful in calculus for writing the equation of a tangent line, particularly because once you know the point of tangency and the derivative (the slope) at that point, writing the equation becomes straightforward.
Let's take the problem example, where the slope \( m = 23 \) and the tangent point is \( (2, 18) \), then applying the point-slope formula gives:\[ y - 18 = 23(x - 2) \]
This allows students to quickly find linear approximations for curved functions such as cubic functions, using tangents. The simplification of this into the slope-intercept form, \( y = 23x - 28 \), also helps and provides an easy path to graph the tangent alongside the curve.