Math Problem Statement
Solution
We are tasked with using a tangent plane to approximate the value of the function at the point . To do this, we need to use the linearization formula, which is the equation of the tangent plane:
where:
- is the point of tangency, here ,
- and are the partial derivatives of evaluated at ,
- will give us the linear approximation of near .
Step 1: Compute and
The function , so its partial derivatives are:
-
Partial derivative with respect to : Applying the chain rule:
-
Partial derivative with respect to : Applying the chain rule:
Step 2: Evaluate and
Now we evaluate the partial derivatives at the point .
Let me calculate these values.We have evaluated the following at the point :
Step 3: Apply the linearization formula
The equation of the tangent plane (linearization) is:
Substituting the values we computed:
Step 4: Approximate
We now approximate the function value at using the linearization:
Let me calculate this next.The linear approximation of the function at the point is approximately , accurate to four decimal places.
Would you like more details or have any questions?
Here are some related questions to explore:
- How do we interpret the meaning of a tangent plane in this context?
- How would the approximation change if we used a different nearby point, such as ?
- What are other ways to approximate the value of a function near a point besides using a tangent plane?
- How would the result differ if we considered a function with higher degrees of nonlinearity?
- Can we find the exact value of and compare it with this approximation?
Tip: The tangent plane is a powerful tool for approximating the behavior of a function near a given point, especially when dealing with smooth surfaces.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Partial Derivatives
Linear Approximation
Formulas
Tangent plane equation: L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
Partial derivatives: f_x(x, y) and f_y(x, y)
Theorems
Chain Rule
Linearization in Multivariable Calculus
Suitable Grade Level
Undergraduate Calculus
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