Step 1: Find the derivative of the curve.
The equation of the curve is:
The derivative of , which gives the slope of the tangent, is:
Step 2: Find the slope of the tangent at .
Substitute into the derivative to find the slope of the tangent:
So, the slope of the tangent at is .
Step 3: Find the equation of the tangent.
The equation of the tangent line is given by the point-slope form:
Where , , and . Substituting the values:
Simplifying:
So, the equation of the tangent is:
Step 4: Find the slope of the normal.
The slope of the normal is the negative reciprocal of the slope of the tangent:
Step 5: Find the equation of the normal.
Using the point-slope form for the normal:
Simplifying:
So, the equation of the normal is:
Final Answer: - Equation of the tangent:
- Equation of the normal: