To find the equation of the tangent line to the graph of \( f(x) = (9 - x^2)^{2/3} \) at the point \( (1, 4) \), we will follow these steps:
**Step 1: Find the derivative of \( f(x) \).**
To find the equation of the tangent line, we first need the derivative of the function \( f(x) \) because the derivative at a point gives us the slope of the tangent line at that point.
The function \( f(x) = (9 - x^2)^{2/3} \) can be differentiated using the chain rule as follows:
Let \( u = 9 - x^2 \). Then \( f(x) = u^{2/3} \).
Differentiating \( u \) with respect to \( x \):
\( \frac{du}{dx} = -2x \)
Differentiating \( f(u) = u^{2/3} \) with respect to \( u \):
\( \frac{df}{du} = \frac{2}{3}u^{-1/3} \)
Using the chain rule for differentiation:
\( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \)
\( \frac{df}{dx} = \frac{2}{3}u^{-1/3} \cdot (-2x) \)
\( \frac{df}{dx} = -\frac{4x}{3} \cdot (9 - x^2)^{-1/3} \)
**Step 2: Calculate the slope at the point \( (1, 4) \).**
To find the slope of the tangent line at \( (1, 4) \), we'll plug in \( x = 1 \) into the derivative:
\( \frac{df}{dx} \bigg|_{x=1} = -\frac{4(1)}{3} \cdot (9 - 1^2)^{-1/3} \)
\( \frac{df}{dx} \bigg|_{x=1} = -\frac{4}{3} \cdot (8)^{-1/3} \)
\( \frac{df}{dx} \bigg|_{x=1} = -\frac{4}{3} \cdot 2^{-1} \) (since \( 8 = 2^3 \))
\( \frac{df}{dx} \bigg|_{x=1} = -\frac{4}{3} \cdot \frac{1}{2} \)
\( \frac{df}{dx} \bigg|_{x=1} = -\frac{4}{6} \)
\( \frac{df}{dx} \bigg|_{x=1} = -\frac{2}{3} \)
So, the slope of the tangent line \( m \) at the point \( (1, 4) \) is \( -\frac{2}{3} \).
**Step 3: Use the point-slope form to write the equation of the tangent line.**
We have a point \( (1, 4) \) and a slope \( -\frac{2}{3} \). The point-slope form of the equation of a line is given by:
\( y - y_1 = m(x - x_1) \)
Here, \( (x_1, y_1) \) is the point on the line, \( m \) is the slope, and \( (x, y) \) are the coordinates of any point on the line. Plugging in the values:
\( y - 4 = -\frac{2}{3}(x - 1) \)
**Step 4: Simplify the equation.**
Expanding the right side:
\( y - 4 = -\frac{2}{3}x + \frac{2}{3} \)
Adding 4 to both sides:
\( y = -\frac{2}{3}x + \frac{2}{3} + 4 \)
\( y = -\frac{2}{3}x + \frac{2}{3} + \frac{12}{3} \)
\( y = -\frac{2}{3}x + \frac{14}{3} \)
The equation of the tangent line at \( (1, 4) \) to the graph \( f(x) = (9 - x^2)^{2/3} \) is:
\( y = -\frac{2}{3}x + \frac{14}{3} \)
**Graphing Note:**
To graph the function \( f(x) = (9 - x^2)^{2/3} \) and the tangent line \( y = -\frac{2}{3}x + \frac{14}{3} \), you can use graphing utilities such as a graphing calculator or graphing software. The function is a transformed and truncated parabola, while the tangent line is a straight line that just touches the curve at the point \( (1, 4) \).