Problem 28 - Calculus: Early Transcendentals

Question

If \( g(x) = x^4 - 2 \), find \( g'(1) \) and use it to find an equation of the tangent line to the curve \( y = x^4 - 2 \) at the point \( (1, -1) \).

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = 4x - 5 \).

Step 1: Find the derivative of g(x)

To find the equation of the tangent line, we first need to find the derivative of the function. The function is given as \( g(x) = x^4 - 2 \). Using the power rule, the derivative is \( g'(x) = 4x^3 \).

Step 2: Evaluate the derivative at x = 1

Now we substitute \( x = 1 \) into the derivative to find the slope of the tangent line. This gives us \( g'(1) = 4(1)^3 = 4 \). Thus, the slope of the tangent line at \( x = 1 \) is 4.

Step 3: Write the point-slope form of the tangent line equation

To find the equation of the tangent line, we use the point-slope form \( y - y_1 = m(x - x_1) \). We know that \( m = 4 \), \( x_1 = 1 \), and \( y_1 = g(1)\). Calculating \( g(1) \), we find \( g(1) = 1^4 - 2 = -1 \). Therefore, the equation becomes \( y + 1 = 4(x - 1) \).

Step 4: Simplify the equation of the tangent line

Simplify the equation \( y + 1 = 4(x - 1) \) by distributing the 4 and solving for \( y \). This gives \( y + 1 = 4x - 4 \). Subtracting 1 from both sides, we arrive at \( y = 4x - 5 \). This is the equation of the tangent line at the point \( (1, -1) \).

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point without crossing it. This unique point is where the tangent line and the curve have the same slope. The tangent line gives a linear approximation of the curve near that point, providing a good understanding of the curve's behavior in a small interval around it. For a function given by a curve, the tangent line represents the best straight-line approximation to the curve close to the point of tangency. The concept is critical in calculus as it helps us understand instantaneous rates of change by approximating non-linear functions linearly.

Power Rule

The power rule is an essential tool in calculus used to find the derivative of functions that are powers of a variable. If you have a function of the form \( f(x) = x^n \), the power rule states that its derivative is \( f'(x) = nx^{n-1} \). This rule simplifies the process of differentiation and is particularly useful when dealing with polynomials. In our case, for \( g(x) = x^4 - 2 \), applying the power rule results in \( g'(x) = 4x^3 \), directly providing a means to determine the slope of the tangent line at any point \( x \). Keep in mind that constants like -2 vanish in differentiation.

Point-Slope Form

The point-slope form is a method used to write the equation of a line when you are given a point on the line and the slope. The form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is the known point on the line. In the context of finding a tangent line, once we have calculated the slope using the derivative (like \( g'(1) = 4 \)), and we know the point \( (1, -1) \) is on the curve, we use these values in the point-slope formula to construct the equation: \( y + 1 = 4(x - 1) \). This format is very straightforward and allows for easy transformation into other forms, such as the slope-intercept form by solving for \( y \).

Slope of Tangent Line

The slope of the tangent line is determined by the derivative of the function evaluated at the particular point where the tangent is drawn. For a function \( y = f(x) \), the derivative \( f'(x) \) represents the rate at which \( y \) changes with respect to a small change in \( x \). In our exercise, to find the slope of the tangent line to the curve \( y = x^4 - 2 \) at the point \( (1, -1) \), we first calculate the derivative to be \( g'(x) = 4x^3 \). We then evaluate it at \( x = 1 \), which gives us \( g'(1) = 4 \). This number, 4, signifies the steepness of the tangent line at that specific point, and determines how the line approximates the curve's appearance there.