How do you sketch the curve #y=e^xsinx# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
Identify those critical points from the first and second derivatives of the function. Then you can "sketch" the curve knowing the basic shape.
See the following reference for details in calculation and interpretation: Weisstein, Eric W. "First Derivative Test." From MathWorldA Wolfram Web Resource. https://tutor.hix.ai Weisstein, Eric W. "Second Derivative Test." From MathWorldA Wolfram Web Resource. https://tutor.hix.ai
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To sketch the curve (y = e^x  \sin(x)) and identify its local maximum, minimum, inflection points, asymptotes, and intercepts:

Intercepts:
 (y)intercept: Set (x = 0) to find (y).
 (x)intercepts: Solve (e^x  \sin(x) = 0) for (x).

Asymptotes:
 Vertical asymptotes: None.
 Horizontal asymptotes: None.

Local Maximum and Minimum:
 Differentiate (y) with respect to (x) to find critical points.
 Use the first derivative test to determine local extrema.

Inflection Points:
 Differentiate (y) twice with respect to (x) to find the second derivative.
 Set the second derivative equal to zero to find points of inflection.
 Determine the concavity of the curve around these points.

Sketch the Curve:
 Plot the intercepts, asymptotes (if any), and critical points.
 Determine the behavior of the curve between critical points using the first derivative test.
 Determine the concavity of the curve using the second derivative test around inflection points.
 Sketch the curve accordingly, ensuring it passes through critical points and behaves as determined by the tests.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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 Is #f(x)=x^44x^3+x4# concave or convex at #x=1#?
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