Use Linear Approximation to estimate the value of f(-1.1, 1.9) when f(-1, 2) = 2 and f_(x)(-1,2)...

Question:

Use Linear Approximation to estimate the value of {eq}f(-1.1, 1.9) {/eq}

when

{eq}f(-1, 2) = 2 {/eq}

and

{eq}f_{x}(-1,2) = -2, f_{y}(-1,2) = 1 {/eq}.

a. 1.7

b. 2

c. 2.1

d. 1.9

e. 1.8

Linear Approximate Value:

To find the approximate value of a function, f(x,y) in the vicinity of a point, we need to use the equation of a tangent plane at the point. A linear approximation is nothing but the equation of a tangent plane at a point on a three-dimensional surface. So, if we have the coordinates of a point and the partial derivatives at the point, we can find the equation of linear approximation. The analogy of the following formula can be applied to find the linear approximation.

{eq}\begin{align} L(x, y)- f(a, b) &= f_{x}(a, b) (x-a)+ f_{y}(a, b) (y-b) & \left[\text{ This equation gives the approximate value of a function about the point }\; (a, b, f(a, b)) \right]\\ \end{align} {/eq}

Answer and Explanation:

{eq}\begin{align} f(-1, 2) &= 2\\ f_{x}(-1,2) & = -2\\ f_{y}(-1,2) &= 1\\ L(x, y)- f(a, b) &= f_{x}(a, b) (x-a)+ f_{y}(a, b) (y-b) & \left[\text{ Apply this formula to find the equation of linear approximation: }\; \right]\\ \Rightarrow L(x, y)- 2 &= (-2) (x+1)+ (1) (y-2) \\ \Rightarrow L(x, y) &= -2 (x+1)+ y-2+2 & \left[\text{ Apply the values } \right]\\ \Rightarrow L(x, y) &= -2 (x+1)+ y & \left[\text{ This is the equation of linear approximation about the point }\; (-1,2, f(-1,2)) \right]\\ \Rightarrow f(-1.1, 1.9) &= L(-1.1, 1.9)\\ &= -2 (-1.1+1)+ 1.9\\ &= -2 (-0.1)+ 1.9\\ &= 0.2 + 1.9\\ &= 2.1 \\ \end{align} {/eq}

Therefore, the estimated value is:

{eq}\displaystyle \boxed{\color{blue} { \matrix{ \begin{align} f(-1.1, 1.9) = 2.1 \\ \text{Correct Answer is represented by} \; :c \\ \end{align}} }} {/eq}


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How to Estimate Function Values Using Linearization

from Math 104: Calculus

Chapter 10 / Lesson 2
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