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AP Calculus BC Challenge Problems

Mr. Shay's Class

You and your partner are to solve a minimum of 17 of these ""fun“ challenge problems: at least 10 easy, at least 5 medium, at least 2 hard. As usual, students wishing to receive exceptional grades must go above and beyond the minimum possible. You are to work only with your partner; do not consult other students or teams, and do not consult WolframAlpha or similar websites. You may ask Mr. Shay for hints beyond the ones provided. Even if you do not completely solve a problem, please provide any work toward the solution. Include full solutions to all completed problems, and write up your solutions as neatly as possible. Credit will be given to both complete and incomplete solutions.

Due Date: Wednesday, May $\mathrm{3}{\mathrm{0}}^{\mathrm{t}\mathrm{h}}$, 2012

1 Easy Problems

1. Let $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ be a one-to-one continuous function such that $\mathit{f}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathrm{4}$ and $\mathit{f}\mathrm{\left(}\mathrm{6}\mathrm{\right)}\mathrm{=}\mathrm{2}$. Assume ${\int }_{\mathrm{1}}^{\mathrm{6}}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{=}\mathrm{1}\mathrm{5}\mathrm{.}$ Calculate ${\int }_{\mathrm{2}}^{\mathrm{4}}{\mathit{f}}^{\mathrm{-}\mathrm{1}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{.}$

2. Find $\mathrm{lim}\underset{\mathrm{‾}}{\mathrm{-}\mathrm{1}\mathrm{+}\mathrm{cos}\mathit{x}}$

$\mathit{x}\mathrm{\to }\mathrm{0}\mathrm{3}{\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathrm{4}{\mathit{x}}^{\mathrm{3}}\mathrm{.}$

3. Find $\underset{\mathit{x}\mathrm{\to }\mathrm{\infty }}{\mathrm{lim}}\sqrt[\mathrm{3}]{{\mathit{x}}^{\mathrm{3}}\mathrm{+}{\mathit{x}}^{\mathrm{2}}}\mathrm{-}\sqrt[\mathrm{3}]{{\mathit{x}}^{\mathrm{3}}\mathrm{-}{\mathit{x}}^{\mathrm{2}}}\mathrm{.}$

4. Find the slope of the tangent line at the point of inflection of $\mathit{y}\mathrm{=}{\mathit{x}}^{\mathrm{3}}\mathrm{-}\mathrm{9}{\mathit{x}}^{\mathrm{2}}\mathrm{-}\mathrm{1}\mathrm{5}\mathit{x}\mathrm{+}\mathrm{3}\mathrm{9}\mathrm{.}$

5. A line through the origin is tangent to $\mathit{y}\mathrm{=}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{3}\mathit{x}\mathrm{+}\mathrm{1}$ at the point $\mathrm{\left(}\mathit{a}\mathrm{,}\mathrm{}\mathit{b}\mathrm{\right)}$ . What is $\mathit{a}$?

6. An object moves along the $\mathit{x}$-axis with its position at any given time $\mathit{t}\mathrm{\ge }\mathrm{0}$ given by $\mathit{x}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{5}{\mathit{t}}^{\mathrm{4}}\mathrm{-}{\mathit{t}}^{\mathrm{5}}\mathrm{.}$ During what time interval is the object slowing down?

7. What is the area of the largest trapezoid that can be inscribed in a semicircle with radius 4 if one of the trapezoid's bases is on the diameter?

8. The highway department of North Eulerina plans to construct a new road between towns Alpha and Beta. Town Alpha lies on a long abandoned road running east-west. Town Beta lies 3 miles north and 5 miles east of Alpha. Instead of building a road directly between Alpha and Beta, the department proposes renovating part of the abandoned road (from Alpha to some point $\mathit{P}$) and then building a new road from $\mathit{P}$ to Beta. If the cost of restoring each mile of old road is \$200, 000 and the cost per mile of a new road is \$400, 000, how much of the old road should be restored in order to minimize cost?

9. Evaluate $\underset{\mathit{n}\mathrm{\to }\mathrm{\infty }}{\mathrm{lim}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{n}}\frac{\mathrm{3}}{\mathit{n}}\sqrt{\mathrm{1}\mathrm{+}\frac{\mathrm{3}\mathit{k}}{\mathit{n}}}\mathrm{.}$

10. Evaluate ${\int }_{\mathrm{0}}^{\mathrm{2}}\frac{\mathit{d}}{\mathit{d}\mathit{x}}\mathrm{\left(}\frac{\mathrm{ln}\mathrm{\left(}\mathit{x}\mathrm{+}\mathrm{1}\mathrm{\right)}}{{\mathit{x}}^{\mathrm{3}}}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{.}$

11. Compute ${\int }_{\mathrm{0}}^{\mathrm{1}}{\mathrm{tan}}^{\mathrm{-}\mathrm{1}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{.}$

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12. Consider the function $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}{\mathit{x}}^{\mathrm{3}}$ and a point $\mathrm{\left(}\mathit{a}\mathrm{,}\mathrm{}{\mathit{a}}^{\mathrm{3}}\mathrm{\right)}$ in the first quadrant. Let $\mathit{A}$ be the area bounded by the $\mathit{y}$-axis, the function $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ , and $\mathit{y}\mathrm{=}{\mathit{a}}^{\mathrm{3}}$, and let $\mathit{B}$ be the area bounded by the $\mathit{x}$-axis, the function $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ and $\mathit{y}\mathrm{=}\mathit{a}$. Find the ratio of the areas $\mathit{A}$: $\mathit{B}\mathrm{.}$

13. What is the volume when the region in the first quadrant bounded by $\mathit{y}\mathrm{=}{\mathit{x}}^{\mathrm{3}}$ and $\mathit{x}\mathrm{=}{\mathit{y}}^{\mathrm{3}}$ is rotated around the $\mathit{y}$-axis?

14. Evaluate $\int \frac{\mathrm{1}}{{\mathit{e}}^{\mathrm{2}\mathit{x}}\mathrm{+}\mathrm{3}{\mathit{e}}^{\mathit{x}}\mathrm{+}\mathrm{2}}\mathit{d}\mathit{x}\mathrm{.}$

15. What is ${\mathit{e}}^{\mathrm{1}}\mathrm{\cdot }{\mathit{e}}^{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{2}}\mathrm{\cdot }{\mathit{e}}^{\mathrm{1}\mathrm{/}\mathrm{3}}\mathrm{\cdot }{\mathit{e}}^{\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{4}}\mathrm{\text{...}}$ ?

2 Medium Problems

1. A continuous real function $\mathit{f}$ satisfies the identity $\mathit{f}\mathrm{\left(}\mathrm{2}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{3}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ for all $\mathit{x}$. If ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{=}\mathrm{1}$, what is ${\int }_{\mathrm{1}}^{\mathrm{2}}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}$?

2. A hallway of width 6 feet meets a hallway of width $\mathrm{6}\sqrt{\mathrm{5}}$ feet at right angles. Find the length of the longest pipe that can be carried horizontally around this corner.

3. What is the area of the largest rectangle that can be drawn inside of a 3-4-5 right triangle with one of the rectangle's sides along the hypotenuse of the triangle?

4. Evaluate ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{ln}\mathrm{\left(}\sqrt{\mathrm{1}\mathrm{-}\mathit{x}}\mathrm{+}\sqrt{\mathrm{1}\mathrm{+}\mathit{x}}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{.}$

5. Find all $\mathit{a}\mathrm{>}\mathrm{1}$ such that ${\int }_{\mathrm{1}}^{\mathit{a}}\mathit{x}\mathrm{ln}\mathit{x}\mathit{d}\mathit{x}\mathrm{=}\frac{\mathrm{1}}{\mathrm{4}}\mathrm{.}$

6. Find the area restricted by the functions $\mathit{y}\mathrm{=}\mathrm{sin}\mathit{x}\mathrm{,}$ $\mathit{x}\mathrm{=}\mathrm{sin}\mathit{y}$, and $\mathit{y}\mathrm{=}\mathit{x}\mathrm{+}\mathrm{2}\mathit{\pi }\mathrm{.}$

7. What is the surface area of the portion of a unit sphere centered at the origin between the planes $\mathit{z}\mathrm{=}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}$ and $\mathit{z}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}$?

8. Discuss the convergence of ${\int }_{\mathrm{-}\mathrm{\infty }}^{\mathrm{0}}{\mathit{e}}^{{\mathit{e}}^{\mathit{x}}\mathrm{+}\mathit{x}}\mathit{d}\mathit{x}$ and if possible, evaluate.

9. What is the fifth derivative of ${\mathit{e}}^{\mathrm{2}\mathit{x}}\mathrm{sin}\mathrm{\left(}\mathrm{x}\mathrm{\right)}$ at $\mathit{x}\mathrm{=}\mathrm{0}$?

10. Find the $\mathrm{2}\mathrm{0}\mathrm{1}{\mathrm{2}}^{\mathrm{t}\mathrm{h}}$ nonzero term of the power series for $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathit{x}}{\mathrm{\left(}{\mathit{x}}^{\mathrm{2}}\mathrm{-}\mathrm{1}{\mathrm{\right)}}^{\mathrm{2}}}$ expanding about $\mathit{x}\mathrm{=}\mathrm{0}\mathrm{.}$

3 Hard Problems

1. Evaluate $\underset{\mathit{x}\mathrm{\to }\mathrm{0}}{\mathrm{lim}}\frac{\mathit{x}\mathrm{tan}\mathrm{\left(}\mathrm{sin}\mathrm{\left(}{\mathit{x}}^{\mathrm{2}}\mathrm{\right)}\mathrm{-}\mathrm{sin}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\right)}}{\mathrm{sin}\mathrm{\left(}{\mathit{x}}^{\mathrm{2}}\mathrm{\right)}}\mathrm{.}$

2. If $\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{,}$ $\mathit{z}$ are real numbers such that $\mathrm{\left(}\mathit{x}\mathrm{+}\mathrm{2}\mathit{y}\mathrm{+}{\mathit{z}}^{\mathrm{2}}{\mathrm{\right)}}^{\mathrm{2}}\mathrm{=}\mathrm{2}\mathit{x}\mathrm{-}\mathit{y}\mathrm{-}{\mathit{z}}^{\mathrm{2}}$, what is the maximum value of $\mathit{y}$? 3. Evaluate ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\frac{\mathrm{ln}\mathit{x}}{{\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathit{x}\mathrm{+}\mathrm{1}}\mathit{d}\mathit{x}\mathrm{.}$

4. Divide a given line segment into two other line segments. Then, cut each of these new line segments into two more line segments. What is the probability that the resulting four line segments are the sides of a quadrilateral?

5. If ${\mathit{a}}_{\mathit{n}}\mathrm{,}$ ${\mathit{b}}_{\mathit{n}}$ are sequences defined as such: ${\mathit{a}}_{\mathit{n}\mathrm{+}\mathrm{1}}\mathrm{=}\frac{\mathrm{1}\mathrm{+}{\mathit{a}}_{\mathit{n}}\mathrm{+}{\mathit{a}}_{\mathit{n}}\mathrm{\cdot }{\mathit{b}}_{\mathit{n}}}{{\mathit{b}}_{\mathit{n}}}\mathrm{,}$ ${\mathit{b}}_{\mathit{n}\mathrm{+}\mathrm{1}}\mathrm{=}\frac{\mathrm{1}\mathrm{+}{\mathit{b}}_{\mathit{n}}\mathrm{+}{\mathit{a}}_{\mathit{n}}\mathrm{\cdot }{\mathit{b}}_{\mathit{n}}}{{\mathit{a}}_{\mathit{n}}}$, with ${\mathit{a}}_{\mathrm{1}}\mathrm{=}\mathrm{1}\mathrm{,}$ ${\mathit{b}}_{\mathrm{1}}\mathrm{=}\mathrm{2}$, find $\underset{\mathit{n}\mathrm{\to }\mathrm{\infty }}{\mathrm{lim}}{\mathit{a}}_{\mathit{n}}\mathrm{.}$

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