Consider the function g defined by g(x) = et - 7x, xER.
Show that the equation of the tangent to the graph of g at x = In 3 may be written in the form
y = - 4x -3(In 3-1).

 

To solve this problem, we need to follow these steps:

### Step 1: Differentiate the Function
Given the function \( g(x) = e^x - 7x \), we need to find the derivative \( g'(x) \), which represents the gradient of the tangent to the curve at any point \( x \).

\[
g'(x) = \frac{d}{dx}(e^x - 7x)
\]
\[
g'(x) = e^x - 7
\]

### Step 2: Find the Gradient at \( x = \ln 3 \)
Now, substitute \( x = \ln 3 \) into the derivative to find the gradient of the tangent line at this point.

\[
g'(\ln 3) = e^{\ln 3} - 7
\]
\[
g'(\ln 3) = 3 - 7 = -4
\]

So, the gradient of the tangent at \( x = \ln 3 \) is \( -4 \).

### Step 3: Find the Coordinates of the Point on the Curve
Next, we need to find the coordinates of the point \( (x_0, y_0) \) on the curve where \( x = \ln 3 \).

\[
y_0 = g(\ln 3) = e^{\ln 3} - 7(\ln 3)
\]
\[
y_0 = 3 - 7\ln 3
\]

Thus, the point of tangency is \( (\ln 3, 3 - 7\ln 3) \).

### Step 4: Write the Equation of the Tangent Line
The equation of a line in point-slope form is:

\[
y - y_0 = m(x - x_0)
\]

Substituting \( m = -4 \), \( x_0 = \ln 3 \), and \( y_0 = 3 - 7\ln 3 \), we get:

\[
y - (3 - 7\ln 3) = -4(x - \ln 3)
\]

Simplify this to get the equation in the required form:

\[
y = -4x + 4\ln 3 + 3 - 7\ln 3
\]

Combine the like terms:

\[
y = -4x + (4\ln 3 - 7\ln 3 + 3)
\]

\[
y = -4x - 3(\ln 3 - 1)
\]

### Final Answer
The equation of the tangent to the graph of \( g \) at \( x = \ln 3 \) can indeed be written as:

\[
y = -4x - 3(\ln 3 - 1)
\]

Would you like more details or have any questions?

### Relative Questions:
1. How do we determine the gradient of a curve at a specific point?
2. What is the significance of the point-slope form of a linear equation?
3. How does the natural logarithm function relate to exponentiation?
4. What is the geometric interpretation of the derivative at a point?
5. Why do we combine like terms when writing the equation of a tangent?

### Tip:
When finding the equation of a tangent line, always calculate the gradient (derivative) at the specific point and use the point-slope form of the equation to simplify the process.

Learn how to find the equation of the tangent line to the graph of g(x) = e^x - 7x at x = ln 3. This problem involves using the derivative of an exponential function and the point-slope form of a line. Suitable for advanced high school students and calculus enthusiasts.

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