Q. 32 In Exercises 31鈥�34, find the p... [FREE SOLUTION]

Chapter 3: Q. 32 (page 288)

In Exercises 31鈥�34, find the point on the graph of the function f that is closest to the point (a, b) by minimizing the square of the distance from the graph to the point.
$f (x) = x^{2}$ and the point $(0, 3)$

Short Answer

Expert verified

The point on the graph of the function that is close to the point is $(\sqrt{\frac{5}{2}}, \frac{5}{2}), (- \sqrt{\frac{5}{2}}, \frac{5}{2})$

Step by step solution

Step 1. Given Information.

The function:

$f (x) = x^{2}$

The point:

$(0, 3)$

Step 2. Write the distance between thee function and the point.

The distance between $(0, 3)$ and $(x, y)$ is:

$D (x) = \sqrt{{(3 - y)}^{2} + {(0 - x)}^{2}} = \sqrt{{(3 - x^{2})}^{2} + {(- x)}^{2}} = \sqrt{x^{4} - 5 x^{2} + 9}$

Step 3. Find the derivative of the function.

$D (x) = \sqrt{x^{4} - 5 x^{2} + 9} D' (x) = \frac{1}{2} {(x^{4} - 5 x^{2} + 9)}^{\frac{- 1}{2}} . (4 x^{3} - 10 x) = \frac{2 (2 x^{3} - 5 x)}{2 (x^{4} - 5 x^{2} + 9)} = \frac{(2 x^{3} - 5 x)}{x^{4} - 5 x^{2} + 9}$

Step 4. Find the point close to the function and the point.

To find the point that is closest to the function and the point,

$D' (x) = 0 \frac{(2 x^{3} - 5 x)}{x^{4} - 5 x^{2} + 9} = 0 2 x^{3} - 5 x = 0 2 x^{2} - 5 = 0 2 x^{2} = 5 x^{2} = \frac{5}{2} x = 卤 \sqrt{\frac{5}{2}}$

Step 5. Find y.

Substitute the value of x in the function to get y,

$y = x^{2} = {(卤 \sqrt{\frac{5}{2}})}^{2} = \frac{5}{2}$

So the closest point is $(\sqrt{\frac{5}{2}}, \frac{5}{2}), (- \sqrt{\frac{5}{2}}, \frac{5}{2})$

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91影视

In Exercises 31鈥�34, find the point on the graph of the function f that is closest to the point (a, b) by minimizing the square of the distance from the graph to the point.
$f (x) = x^{2}$ and the point $(0, 3)$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Write the distance between thee function and the point.

Step 3. Find the derivative of the function.

Step 4. Find the point close to the function and the point.

Step 5. Find y.

One App. One Place for Learning.

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91影视

In Exercises 31鈥�34, find the point on the graph of the function f that is closest to the point (a, b) by minimizing the square of the distance from the graph to the point.f(x)=x2 and the point (0,3)

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Write the distance between thee function and the point.

Step 3. Find the derivative of the function.

Step 4. Find the point close to the function and the point.

Step 5. Find y.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Pure Maths

Applied Mathematics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 31鈥�34, find the point on the graph of the function f that is closest to the point (a, b) by minimizing the square of the distance from the graph to the point.
$f (x) = x^{2}$ and the point $(0, 3)$