Problem 45
Question
In Exercises 43-50, (a) find the slope of the graph of \(f\) at the given point, (b) use the result of part (a) to find an equation of the tangent line to the graph at the point, and (c) graph the function and the tangent line. \(f(x)= x^3 - 2x, \quad (1, -1)\)
Step-by-Step Solution
Verified Answer
The slope of the tangent line to the function \(f(x) = x^3 - 2x\) at the point (1, -1) is 1 and the equation of this tangent line is \(y = x - 2\).
1Step 1: Find the Derivative
The first step is to find the derivative of the function \(f(x) = x^3 - 2x\) \(f'(x) = 3x^2 - 2\)
2Step 2: Find the Slope at Given Point
We can compute the slope of the function at the given point (1, -1) by substituting 1 into our derivative. \[f'(1) = 3*1^2 - 2 = 1. \] So the slope of the tangent line to the function at the point (1, -1) is 1.
3Step 3: Find the Equation of the Tangent Line
The equation of the line tangent to the function at the point (1, -1) can be represented in point-slope form \(y - y1 = m(x - x1)\), where m is the slope and \((x1, y1)\) is the point on the line. Here, \(m = 1\) and \((x1, y1) = (1, -1)\). So the equation of the tangent line is: \(y - (-1) = 1 * (x - 1)\), which simplifies to \(y = x - 2.\)
4Step 4: Graph the Function and the Tangent Line
The original function \(f(x) = x^3 - 2x\) is a cubic function and its graph has a typical cubic shape. The tangent line is a straight line \(y = x - 2\) and passes through the point \((1, -1)\) on the graph of the function with a slope of 1. This would normally be done on graph paper or using graphing software.
Key Concepts
DerivativeSlopeCubic Function
Derivative
The concept of a derivative is fundamental in calculus and plays a key role in understanding how functions behave. The derivative of a function gives us the rate at which the function’s value changes with respect to a change in the input value. This is often described as the slope of the tangent line to the graph of the function at a particular point.
When calculating the derivative of a function like \( f(x) = x^3 - 2x \), we are essentially determining the formula that gives us the slope of the tangent line at any point \( x \) on the curve. To find the derivative, we use rules from calculus such as the power rule. Applying these rules:
When calculating the derivative of a function like \( f(x) = x^3 - 2x \), we are essentially determining the formula that gives us the slope of the tangent line at any point \( x \) on the curve. To find the derivative, we use rules from calculus such as the power rule. Applying these rules:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \(-2x\) is \(-2\).
Slope
The slope of a line is a measure of its steepness. In the context of a function's graph, the slope of the tangent line at a particular point is what you obtain from the derivative at that point. It tells you how sharply the function is increasing or decreasing at that instant. For the function \( f(x) = x^3 - 2x \), we found its derivative to be \( f'(x) = 3x^2 - 2 \).
To find the slope at the point \((1, -1)\), we evaluate the derivative at \( x = 1 \). Thus,
To find the slope at the point \((1, -1)\), we evaluate the derivative at \( x = 1 \). Thus,
- \( f'(1) = 3(1)^2 - 2 = 1 \).
Cubic Function
A cubic function is a polynomial of degree three, and it has the general form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants and \( a eq 0 \). Cubic functions like \( f(x) = x^3 - 2x \) have characteristic shapes that can include one or two turning points, creating local maxima and minima.
When analyzing cubic functions, their graphs typically exhibit a property called inflection points. This is where the graph changes concavity, transitioning from being concave up to concave down, or vice versa. For our function, the graph begins like a typical cubic curve but is modified by the \(-2x\) term, affecting its symmetry and how sharply it rises and falls.
The function \( f(x) = x^3 - 2x \) points to a cubic graph where the interesting behavior happens around the origin, as negative and positive \( x \) values impact its steepness and direction differently. Visualizing these properties through graphing allows you to see where the slope (from the derivative) varies and how the tangent lines intersect the curve, thus offering deeper insights into this polynomial's weirdly satisfying symmetry visually.
When analyzing cubic functions, their graphs typically exhibit a property called inflection points. This is where the graph changes concavity, transitioning from being concave up to concave down, or vice versa. For our function, the graph begins like a typical cubic curve but is modified by the \(-2x\) term, affecting its symmetry and how sharply it rises and falls.
The function \( f(x) = x^3 - 2x \) points to a cubic graph where the interesting behavior happens around the origin, as negative and positive \( x \) values impact its steepness and direction differently. Visualizing these properties through graphing allows you to see where the slope (from the derivative) varies and how the tangent lines intersect the curve, thus offering deeper insights into this polynomial's weirdly satisfying symmetry visually.
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