1 Calculus I

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Plot $y^3=6x^2-x^3$ following class procedures¹²³. [1, Maximum and Minimum Values, Curve Sketching]
2. Plot $y=x\sqrt[3]{2-x}$ following class procedures¹². [1, Maximum and Minimum Values, Curve Sketching]
3. Plot $y=\ln(e^{x+1}-1)$ following class procedures¹². [1, Maximum and Minimum Values, Curve Sketching]

¹ Include a plot

² Discuss intercepts, extents, symmetries, asymptotes, asymptotic behavior for $x\to\pm\infty$, local/global maxima/minima, concavity, inflection points, poles, cusps, corners, vertical slopes, and other singularities

³ Note that $\lim_{x\to\infty}=\lim_{u\downarrow0}$ if $u=1/x$. Alternatively, use the Taylor series approximation $(1+\varepsilon)^p\sim1+p\varepsilon$ for $\varepsilon\to0$