Problem 10 - Calculus Early Transcendentals

Question

Equations of tangent lines by definition (1) a. Use definition (1) (p. 127 ) to find the slope of the line tangent to the graph of \(f\) at \(P\) b. Determine an equation of the tangent line at \(P\). c. Plot the graph of \(f\) and the tangent line at \(P\). $$f(x)=-3 x^{2}-5 x+1 ; P(1,-7)$$

Step-by-Step Solution

Answer

Question: Find the slope of the tangent line to the graph of the function \(f(x) = -3x^2 - 5x + 1\) at the point \(P(1, -7)\) and determine the equation of the tangent line. Finally, plot both the graph of the function and the tangent line. Answer: The slope of the tangent line is \(-11\), and the equation of the tangent line is \(y = -11x + 4\). Plot the graph of the function and the tangent line to visualize their relationship at point \(P(1, -7)\).

Step 1: Find the derivative of \(f(x)\)

The first step is to find the derivative of the given function. Since \(f(x) = -3x^2 - 5x + 1\), the derivative \(f'(x)\) can be found as follows: $$f'(x) = \frac{d}{dx}(-3x^2 - 5x + 1) = -6x - 5$$

Step 2: Find the slope of the tangent line

Now that we have found the derivative, we can use it to find the slope of the tangent line at the given point \(P(1, -7)\). To do this, we substitute the x-coordinate of the point into the derivative: $$m = f'(1) = -6(1) - 5 = -11$$

Step 3: Determine the equation of the tangent line

We now have the slope of the tangent line and the point it passes through, so we can use the point-slope form to find the equation of the tangent line: $$y - y_1 = m(x - x_1)$$ $$y - (-7) = -11(x - 1)$$ Simplifying this equation gives us the equation of the tangent line: $$y = -11x + 4$$

Step 4: Plot the graph of \(f(x)\) and the tangent line

It's time to plot the graph of the given function \(f(x) = -3x^2 - 5x + 1\) and the tangent line \(y = -11x + 4\) at the point \(P(1, -7)\). You can use graphing software or graph paper to plot these functions. A visual representation will show the function's parabolic shape and how the tangent line touches the function's graph at the point \(P(1, -7)\).

Key Concepts

Derivative

The derivative of a function is a key concept in calculus, fundamental for finding tangent lines. At its core, the derivative represents the rate at which a function is changing at any given point. Think of it as the slope of the tangent line to the curve of a function at a particular point.
For the function given, \(f(x) = -3x^2 - 5x + 1\), the derivative \(f'(x)\) is

• \(f'(x) = -6x - 5\)

This tells us how steep the function is at any \(x\) value. To find the slope of the tangent line at a specific point, such as \(P(1, -7)\), substitute the \(x\)-coordinate (

• \(f'(1) = -11\)

This gives the slope of the tangent line at \(x = 1\). Calculating derivatives can sometimes be challenging, but it follows basic rules of differentiation, which apply to polynomials: power rule, constant rule, and sum rule. With practice, it becomes a powerful tool in calculus.

Point-Slope Form

The point-slope form is a method to find the equation of a line given a point and the slope. It is expressed as:

• \[ y - y_1 = m(x - x_1) \]

Where \((x_1, y_1)\) is the point on the line and \(m\) is the slope.
In our example, you have the point \((1, -7)\) and the slope \(-11\). Substituting these values into the point-slope form yields:

• \[ y - (-7) = -11(x - 1) \]
• \[ y + 7 = -11x + 11 \]

By simplifying the equation, you arrive at the equation of the line:

• \( y = -11x + 4 \)

This form is particularly useful in finding the equation of the tangent line, which accurately describes the line that just "touches" a curve at a specific point without intersecting it elsewhere.

Graphing Functions

Graphing functions helps visualize how functions and their derivatives behave.
For the quadratic function \(f(x) = -3x^2 - 5x + 1\), its graph is a parabola opening downward due to the negative leading coefficient. The tangent line graph gives a straight line that touches the curve at point \(P(1, -7)\).
To graph these:

• Plot the function \(f(x)\) to showcase the overall shape.
• Draw the tangent line \(y = -11x + 4\).

This line should only touch the curve at \((1, -7)\). The intersection point demonstrates how the tangent line represents the instantaneous rate of change at that point. Visualizing graphs not only helps in understanding the nature of functions but also illustrates the significance of the tangent line as showing the best linear approximation of the curve at a point.