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Section 25 Curve sketching II
Objectives
Bring all our calculus tools to bear on the analysis of a real-valued function \(f\text{.}\) In more detail: determine the domain, the \(y\) -intercept and any \(x\) -intercepts; compute the limit of the function at endpoints of the domain and determine whether there are horizontal asymptotes; determine whether there are vertical asymptotes; find and classify all critical points of \(f\text{;}\) determine the intervals of monotonicity; determine intervals of constant concavity; identify any inflection points.
Express the results of our analysis of a function’s behavior in the form of a detailed sketch.
Example 25.1 . Curve sketch: trig.
Provide a graph of
\(f(x)=\sin x+\sin^2 x\) that includes all the details listed in
Procedure 24.1 to the
best of your abilities .
Solution .
Domain and intercepts.
Endpoint behavior and vertical asymptotes.
Critical points and intervals of monotonicity.
Concavity and inflection points.
Example 25.2 . Curve sketching: rational function.
Provide a graph of
\(f(x)=\frac{\sqrt{x}}{x-1}\) that includes all the details listed in
Procedure 24.1 .
Solution .
Domain and intercepts.
The domain of \(f\) is \([0,1)\cup (1,\infty)\text{.}\) The point \((0,0)\) is both the unique \(x\) -intercept and the \(y\) -intercept.
Endpoint behavior and vertical asymptotes.
Critical points and intervals of monotonicity.
Concavity and inflection points.
