Let f(x, y) = 2x + 3y - 4. Find the slope of the line tangent to this surface at the point (2, -1) and lying in the a. plane x = 2 b. plane y = -1.
Understand the Problem
The question is asking us to find the slope of the line tangent to the surface defined by the function f(x, y) = 2x + 3y - 4 at a specific point (2, -1) and in two different planes: x = 2 and y = -1.
Answer
a. $2$ b. $3$
Answer for screen readers
The slopes of the tangent lines are:
a. For plane $x = 2$: $2$
b. For plane $y = -1$: $3$
Steps to Solve
- Find the partial derivative with respect to x
To find the slope of the tangent line in the plane $x = 2$, we first calculate the partial derivative of the function $f(x, y)$ with respect to $x$.
$$ \frac{\partial f}{\partial x} = 2 $$
- Evaluate the partial derivative at the point (2, -1)
Now we evaluate this partial derivative at the point $(2, -1)$.
$$ \frac{\partial f}{\partial x}(2, -1) = 2 $$
- Find the partial derivative with respect to y
Next, we find the partial derivative of the function $f(x, y)$ with respect to $y$ to determine the slope of the tangent line in the plane $y = -1$.
$$ \frac{\partial f}{\partial y} = 3 $$
- Evaluate the partial derivative at the point (2, -1)
Now we evaluate this partial derivative at the point $(2, -1)$.
$$ \frac{\partial f}{\partial y}(2, -1) = 3 $$
The slopes of the tangent lines are:
a. For plane $x = 2$: $2$
b. For plane $y = -1$: $3$
More Information
This problem involves calculating partial derivatives to find the slopes of tangent lines in specified planes. The interpretation of the slopes is related to how the function changes with respect to each variable independently.
Tips
- Forgetting to evaluate the partial derivatives at the given point.
- Confusing the meanings of the partial derivatives; remember that one is with respect to $x$ and the other is with respect to $y$.
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