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Calculus: Derivatives
Using derivatives in optimization problems is one of the primary goals of
the first semester of calculus. Many of the problems below can be solved
by using the first derivative. However, some of the problems, such as
Making
Money at the Movies, are based on a quadratic function and can be solved
by using non-calculus methods.
A brief explanation of derivatives
and a look at the limit
definition of a derivative can be found in the
High School Calculus area
of the Ask Dr. Math archives. To find relevant sites on the Web, browse and
search Calculus (Single Variable):
Differentiation
in our Internet Mathematics Library.
Access to these problems requires a Membership.
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Another Product Max
- Leigh Nataro
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Trig/Calc, difficulty level 5. Use calculus to find the maximum product for xyz if x + y + z = B, where B is a real number.
... more>>
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Around the Corner
- Louis Holmes
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Trig/Calc, difficulty level 4. A girder must be carried along a corridor that has a 90-degree turn at the end. How wide must the corridor be in order for the girder to fit around the corner?
... more>>
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Bacterial Growth Rates
- Lisa Lavelle
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calcpow, difficulty level 1. Compare the growth rates of two bacteria as time approaches infinity.
... more>>
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Building a Sandbox
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find the maximum area for a triangular-shaped sandbox.
... more>>
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Building Boxes
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the maximum volume of a rectangular prism constructed from an 8.5-inch by 11-inch piece of paper.
... more>>
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Costly Cable
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find where to lay a fiber optic cable in order to minimize the cost.
... more>>
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Environmentally Friendly
- Leigh Nataro
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Trig/Calc, difficulty level 2. What dimensions for a given page will use the least amount of paper?
... more>>
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Find the Function
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find a cubic polynomial that has a given maximum, minimum, and point of inflection.
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The General Box Company
- Leigh Nataro
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Trig/Calc, difficulty level 3. A rectangular piece of cardboard is folded up to make a box. Find a general formula for the maximum volume of the box.
... more>>
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George's Gutters
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the maximum volume of a gutter.
... more>>
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The Indoor Driving Range
- Leigh Nataro
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Trig/Calc, difficulty level 3. Determine the minimum amount of netting needed to make an indoor driving range golf cage.
... more>>
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Is It a Midpoint?
- Leigh Nataro, Andrea Medini
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Trig/Calc, difficulty level 3. Prove that the point of inflection of a cubic function is the midpoint of the segment that connects the function minimum and the function maximum.
... more>>
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Job Interview at the Can-Can Company
- Leigh Nataro
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Trig/Calc, difficulty level 3. Find the smallest amount of tin needed to make a can that can hold 12 cubic inches.
... more>>
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Let the Sunshine In
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the dimensions of a window designed to let in the maximum amount of sunshine.
... more>>
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Making Money at the Movies
- Leigh Nataro
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Trig/Calc, difficulty level 3. How much should the manager of a theater raise his prices in order to maximize his profit?
... more>>
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My Snow-Shoveling Business
- Leigh Nataro
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Trig/Calc, difficulty level 2. Determine the number of customers needed to maximize the revenue for a snow-shoveling business.
... more>>
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Nice Shot, Robin Hood!
- Sara Aucoin, Marcie Sawyer, Pattie Draper
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Trig/Calc, difficulty level 2. Answer some questions about the flight of Robin Hood's arrow.
... more>>
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Pete's Predictions
- Leigh Nataro
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Trig/Calc, difficulty level 2. Verify the formula for finding the maximum height of a ball thrown upward.
... more>>
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Product Max
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the maximum value for two numbers whose sum is a.
... more>>
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Rolle's Theorem Rediscovered
- John Feminella
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Trig/Calc, difficulty level 3. Prove the Mean Value theorem.
... more>>
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Shipping Packages
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the maximum volume of a box based on the given information.
... more>>
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The Super-Ultra-Mega-Big Cup
- Alvirne High School Calculus Class
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Trig/Calc, difficulty level 3. What is the minimum cost to make a cup based on given information?
... more>>
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The Triangle Inside an Angle
- Ramani Jv
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Trig/Calc, difficulty level 5. Based on the given information, find the area of the smallest triangle contained within an angle.
... more>>
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The Unit Circle Triangle
- Leigh Nataro
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Trig/Calc, difficulty level 2. Find the area of the largest right triangle in the first quadrant of the unit circle.
... more>>
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Your Interview Question Is...
- Leigh Nataro
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Trig/Calc, difficulty level 3. If you want to minimize the surface area of a can with a fixed volume, what should be the ratio of the height to the radius?
... more>>
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