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Chapter 3: Problem 12

a. Use definition (1) 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 \(P\). $$f(x)=-3 x^{2}-5 x+1 ; P(1,-7)$$

Short Answer

Expert verified
Answer: The equation of the tangent line is \(y = -11x + 4\).

Step by step solution

01

Find the slope of the tangent line

To find the slope of the tangent line at point \(P\), we will first find the derivative of the function \(f(x)\), which is the slope of the tangent line: $$f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$ Now, substitute the given function and simplify within the limit: $$f'(x) = \lim_{h\to0} \frac{-3 (x + h)^2 - 5(x + h) + 1 - (-3x^2 - 5x + 1)}{h}$$ Expand the numerator and simplify: $$f'(x) = \lim_{h\to0} \frac{-3(x^2 + 2xh + h^2) - 5(x + h) + 1 + 3x^2 + 5x - 1}{h}$$ $$f'(x) = \lim_{h\to0} \frac{-3x^2 - 6xh - 3h^2 - 5x - 5h + 3x^2 + 5x}{h}$$ $$f'(x) = \lim_{h\to0} \frac{- 6xh - 3h^2 - 5h}{h}$$ Now, factor out \(h\) from the numerator and cancel it with the \(h\) in the denominator: $$f'(x) = \lim_{h\to0} (-6x - 3h - 5)$$ Finally, evaluate the limit as \(h\) approaches 0: $$f'(x) = -6x - 5$$ Now, to find the slope of the tangent line at point \(P\), we will evaluate \(f'(x)\) at x = 1: $$m = f'(1) = -6(1) - 5 = -11$$
02

Determine the equation of the tangent line

We know the slope of the tangent line at point \(P(1, -7)\) is -11. We can use the point-slope form to find the equation of the tangent line: $$y - y_1 = m(x - x_1)$$ Here, \(x_1\) and \(y_1\) are the coordinates of the point \(P\). Substitute the found slope and point \(P\) into the point-slope form: $$y - (-7) = -11(x - 1)$$ Now, simplify the equation: $$y + 7 = -11x + 11$$ $$y = -11x + 11 - 7$$ $$y = -11x + 4$$ So, the equation of the tangent line is: \(y = -11x + 4\).
03

Plot the graph of \(f\) and the tangent line

To plot the graph, we will use a graphing tool to visualize the function \(f(x) = -3x^2 - 5x + 1\) and the tangent line \(y = -11x + 4\) with the point \(P(1, -7)\). You can use a graphing calculator or an online plotting tool, such as Desmos or Geogebra, to enter the given functions and the point. The plot will show the function and tangent line intersecting at point \(P\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives Explained
The derivative is a fundamental concept in calculus. It represents the rate at which a function is changing at any given point. In simpler terms, the derivative at a point gives the slope of the tangent line to the function's graph at that point.
The formula to find the derivative of a function, using the definition, is:
  • \(f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}\)
This definition demonstrates how the derivative is effectively the limit of the average rate of change as the interval between two points on the graph approaches zero.
In the exercise given, the derivative of the function \(f(x) = -3x^2 - 5x + 1\) was calculated to find \(f'(x) = -6x - 5\). This expression gives us the slope of the tangent line at any point \(x\) on the function.
Understanding Tangent Lines
A tangent line is a straight line that touches a curve at a single point without crossing over. The line shares the same slope as the curve at the point of tangency. This makes the concept of tangency unique, as the tangent line represents the best linear approximation of the curve at that specific point.
When we need to find the equation of the tangent line, we not only require the point of tangency but also the slope of the tangent line, which is provided by the derivative.
In our exercise, after determining that the slope of the tangent line at \(P(1, -7)\) is \(-11\), we use the point-slope form equation:
  • \(y - y_1 = m(x - x_1)\)
Substituting the given point \(P\) and the slope, we derive the tangent line equation as \(y = -11x + 4\).
Graphing Functions and Tangent Lines
Graphing is a visual representation of mathematical functions. It is essential for understanding how functions behave and how tangent lines intersect them at specific points.
To explore this, imagine plotting the graph of the function \(f(x) = -3x^2 - 5x + 1\). This graph is a parabola that opens downwards.
Using graphing tools like Desmos or GeoGebra can help in visualizing not only this parabola but also how the tangent line \(y = -11x + 4\) intersects it at \(P(1, -7)\).
  • This intersection signifies that the tangent line and the parabola have identical slopes at \(x = 1\).
This overlapping at a single point emphasizes the precision of finding derivatives and tangent lines.
Slope of the Tangent Line
The slope of the tangent line is a significant concept that defines how steep a line is at a specific point. The slope indicates how much the function is increasing or decreasing at that point.
When a derivative \(f'(x)\) is evaluated at a particular \(x\)-value, it gives the slope of the tangent line at that point.
  • For the function \(f(x) = -3x^2 - 5x + 1\), computing the derivative gives us \(f'(x) = -6x - 5\).
Evaluating this at \(x = 1\), we find the slope to be \(-11\).
This means that at \(P(1, -7)\), the function is decreasing rapidly, as indicated by the negative slope. The clearer understanding of the slope is crucial for comprehending the behavior of curves and their tangent lines.

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Most popular questions from this chapter

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