dy = ². The graph of f passes through the point dx Let f be a function such that at each point (x, y) on the graph of f, the slope is given by (1, -2) and is concave up on the interval 1 < x < 1.5. Let k be the approximation for f (1.3) found by using the locally linear approximation of f at x = 1. Which of the following statements about k is true? k = -2.65 and is an underestimate for f (1.3). A B (0) C D k = -2.65 and is an overestimate for f (1.3). k = -2.15 and is an underestimate for f (1.3). k = -2.15 and is an overestimate for f (1.3).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
Let \( f \) be a function such that at each point \((x, y)\) on the graph of \( f \), the slope is given by 

\[
\frac{dy}{dx} = \frac{1}{2}x - \frac{1}{4}y^2.
\]

The graph of \( f \) passes through the point \((1, -2)\) and is concave up on the interval \( 1 < x < 1.5 \). Let \( k \) be the approximation for \( f(1.3) \) found by using the locally linear approximation of \( f \) at \( x = 1 \). Which of the following statements about \( k \) is true?

- (A) \( k = -2.65 \) and is an underestimate for \( f(1.3) \).
- (B) \( k = -2.65 \) and is an overestimate for \( f(1.3) \).
- (C) \( k = -2.15 \) and is an underestimate for \( f(1.3) \).
- (D) \( k = -2.15 \) and is an overestimate for \( f(1.3) \).
Transcribed Image Text:Let \( f \) be a function such that at each point \((x, y)\) on the graph of \( f \), the slope is given by \[ \frac{dy}{dx} = \frac{1}{2}x - \frac{1}{4}y^2. \] The graph of \( f \) passes through the point \((1, -2)\) and is concave up on the interval \( 1 < x < 1.5 \). Let \( k \) be the approximation for \( f(1.3) \) found by using the locally linear approximation of \( f \) at \( x = 1 \). Which of the following statements about \( k \) is true? - (A) \( k = -2.65 \) and is an underestimate for \( f(1.3) \). - (B) \( k = -2.65 \) and is an overestimate for \( f(1.3) \). - (C) \( k = -2.15 \) and is an underestimate for \( f(1.3) \). - (D) \( k = -2.15 \) and is an overestimate for \( f(1.3) \).
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