Tutorial Week 8

We’ll be focusing on curve sketching using first and second derivatives.

The curve sketching guidlines are as follows:

1. Domain

2. Intercepts (x and y intercepts)

3. Symmetry (odd or even)

4. Asymptotes (horizontal, vertical, slant/oblique)

5. Critical points (and intervals of increase/decrease)

6. Concavity (and inflection points)

7. Sketch

Curve Sketching

Q1: Sketch \(f(x) = \ln\left(x\right)^{2}\).

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Q2: Sketch \(f(x) = x\sqrt{2-x^{2}}\). Where are the local minimums or local maximums?

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Sinusodial Function Derivatives

Q3: Find the local maximums and minimums of \(f(x) = \csc\left(x\right)+\sin x\).

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Test for Absolute Extrema

Q4: Let \(f(x) = \frac{1}{7}\left(x^{3}-2x^{2}+x+2\right)\) represent the water level of a lake x hours after 9am. Between 9am and 1pm (inclusive), when was the water highest? When was it lowest?

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Q5: Determine the absolute extrema of \(f(x) = 2\sqrt{x}-x\) on the interval \(x \in [0, 9]\).

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Drawing Functions

Q6: Draw a function with domain \([0, 5]\) that does not satisfy the conclusion of the Extreme Value Theorem on the interval \([0, 5]\).

Q7: Draw a function with domain \([0, 5]\) that does not satisfy the conclusion of the Extreme Value Theorem on the interval \([2, 3]\).

Q8: Draw a continuous function with domain \((-2, 2)\) that does not satisfy the conclusion of the Extrema Value Theorem on its domain.