Problem 23 - Calculus Volume 1

Question

Consider the below function, $$ f(x)=x^{2}+9 x, a=2 $$ Find the slope of the tangent line \(f^{\prime}(a)\) and the equation of the tangent line at \(x=a\)

Step-by-Step Solution

Answer

The slope is 13; the tangent line equation is \( y = 13x - 4 \).

Step 1: Find the Derivative

To find the slope of the tangent line at a point, we first need the derivative of the function. Start by differentiating the function \( f(x) = x^2 + 9x \). The derivative, \( f'(x) \), is found using the power rule for each term: \[ f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(9x) = 2x + 9. \]

Step 2: Evaluate the Derivative at x = a

The next step is to evaluate the derivative at \( x = a \) to find the slope of the tangent line at that point. We substitute \( a = 2 \) into the derivative: \[ f'(2) = 2(2) + 9 = 4 + 9 = 13. \] The slope of the tangent line at \( x = 2 \) is 13.

Step 3: Find the Equation of the Tangent Line

To find the equation of the tangent line, we need a point on the line and its slope. We have the slope \( m = 13 \). The point on the curve at \( x = 2 \) is obtained by evaluating the original function at \( x = 2 \): \[ f(2) = (2)^2 + 9(2) = 4 + 18 = 22. \] The point is \((2, 22)\). The equation of the tangent line using the point-slope form is: \[ y - 22 = 13(x - 2). \] Simplifying, this gives: \[ y = 13x - 26 + 22 \] \[ y = 13x - 4. \]

Step 4: Verification

Verify the equation by checking that it satisfies both the slope and point conditions. The slope from the equation \( y = 13x - 4 \) is 13, which matches \( f'(2) = 13 \). The point (2, 22) should lie on this line, and substituting \( x = 2 \) in the tangent equation, \( y = 13(2) - 4 = 22 \) confirms it does.

Key Concepts

Derivatives

A derivative represents how a function changes as its input changes. Think of it as a tool that tells us the rate at which something is changing at any given point. In calculus, derivatives are crucial for understanding how functions behave and finding slopes of curves and other critical points.

• To differentiate a function means to calculate its derivative.
• The derivative of a function at a particular point gives us the slope of the tangent line to the graph of the function at that point.
• We often denote the derivative of a function \( f(x) \) as \( f'(x) \) or \( \frac{df}{dx} \).

In our exercise, to find the derivative of the function \( f(x) = x^2 + 9x \), the power rule is applied to calculate \( f'(x) = 2x + 9 \). Now, with the derivative in hand, we can proceed to evaluate it at specific points to gain more insight into the function's behavior.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing over. This line has the same slope as the curve at that point. Understanding tangent lines can provide insight into the function's behavior at specific points.

• The slope of the tangent line is equal to the derivative of the function at the point of tangency.
• Tangent lines can approximate curves nearby the point of tangency.
• If you know the derivative's value and a point on the curve, you can write the equation for the tangent line.

In step 3 of our exercise, the tangent line is found using the point \((2, 22)\) on the curve \( f(x) \) and the slope derived earlier, which is 13. The equation is crafted using the point-slope form, resulting in \( y = 13x - 4 \). Such a line closely follows the curve near \( x = 2 \).

Power Rule

The power rule is a simple and powerful tool in calculus for finding derivatives. It's incredibly useful for functions consisting of terms like \( x^n \), where \( n \) is a real number. Applying this rule helps streamline the process of differentiation.

• The general power rule is: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• It helps to quickly find the derivative of each term in a polynomial function.
• Combine the derivatives of individual terms to get the derivative of the entire function.

Using the power rule, the derivative of \( x^2 \) becomes \( 2x \) and the derivative of \( 9x \) is simply 9. Thus, \( f'(x) = 2x + 9 \), which we computed earlier in the exercise. The power rule simplifies finding the slope of the tangent line at any point on the function curve.

Slope of a Curve

The slope of a curve at a given point shows how steep the curve is at that location. It is given by the derivative of the function at that point and is crucial for understanding how the curve rises or falls.

• A positive slope indicates a rising curve at that point.
• A negative slope means the curve is falling.
• A zero slope typically indicates a flat point, possibly a local maximum or minimum.

In the problem, finding the slope of the curve at \( x=2 \) involves evaluating the derivative \( f'(2) = 13 \). This tells us that the curve is rising steeply at this point with a slope of 13. Knowing the slope helps in drawing the tangent line accurately, making the slope a key part of curve analysis.