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Sketching

# Sketching - c,f c is neither a relative maximum nor a...

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Sketching the curve of y = f ( x ) 1. Find any x - and y -intercepts. x -intercept is a point where y = 0. y -intercept is a point where x = 0. For rational functions we may consider the vertical or horizontal asymptotes. x = k making the denominator zero is the vertical asymptote. y = lim x →∞ f ( x ) is the horizontal asymptote. 2. Find all critical points for y = f ( x ) that are all values of x where f 0 ( x ) = 0 or where f 0 ( x ) is undeﬁned. 3. Knowing the critical points, determine where the function f ( x ) is increasing or decreasing. f ( x ) is increasing where f 0 ( x ) > 0. f ( x ) is decreasing where f 0 ( x ) < 0. 4. Find the relative maximum and minimum values. First Derivative Test : Let ( c,f ( c )) be a critical point. (a) If f 0 ( x ) changes from positive to negative at x = c , then the point ( c,f ( c )) is a relative maximum. (b) If f 0 ( x ) changes from negative to positive at x = c , then the point ( c,f ( c )) is a relative minimum. (c) If f 0 ( x ) does not change sign at x = c , then the point (

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Unformatted text preview: c,f ( c )) is neither a relative maximum nor a relative minimum. Second Derivative Test : Let ( c,f ( c )) be a critical point. (a) If f 00 ( c ) &amp;lt; 0, then the point ( c,f ( c )) is a relative maximum. (b) If f 00 ( c ) &amp;gt; 0, then the point ( c,f ( c )) is a relative minimum. Note that if f 00 ( c ) = 0 or if f 00 ( c ) is undened, then the test fails and the rst derivative test should be used instead. 1 2 5. Find the points of inection that are all values of x where f 00 ( x ) = 0 and f 00 changes the sign at those points. 6. Knowing the points of inection, determine where the function f ( x ) is concave up or concave down. (a) f ( x ) is concave up where f 00 ( x ) &amp;gt; 0. (b) f ( x ) is concave down where f 00 ( x ) &amp;lt; 0. 7. Sketch the curve using all information....
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Sketching - c,f c is neither a relative maximum nor a...

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