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7 Cards in this Set
- Front
- Back
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How do you find the tangent line at a certain point?
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1. Find derivative of f(x), this is your slope. f ' (x) = m.
2. If m is in terms of x, plug in value of x to find m. 3. Plug m, x1, and y1 (original points) into y - y1 = m(x-x1) 4. Solve for y. |
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What does it mean to find the equation of the line normal to the curve f(x) at a certain point?
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It means to find the tangent line first then utilize the inverse reciprocal of that answer m, which is -1/m, to plug into the equation y - y1 = m(x -x1).
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What is the difference between an absolute max and min and a local max and min?
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An absolute max+min is the largest or smallest value that the function will ever take on the domain. A local max+min is the largest or smallest value in a certain interval, not the endpoints of said interval (a < c < b ).
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How would you find the absolute maximum and absolute minimum of a function?
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1. Understand that if f(c) is a local extreme value and f ' (c) exists then f ' (c) = 0. Note that the converse is false.
2. If f ' (c) = 0 or f ' (c) DNE then c is a critical #. 3. Set f ' (x) = 0 and find critical #s (where x = 0 or DNE). 4. Plug x back into original equation. 5. If closed interval is present, then plug closed #s into original equation as well. 6. Largest # is absolute maximum, smallest # is absolute minimum. |
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How do you find the increasing and decreasing values of a function?
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1. Find critical #s (where f(x) = 0).
2. Create a number line and either plug in points into f ' (x) or use trends to find out if the space between the critical #s are positive or negative. 3. If the gap is positive then the interval is increasing. If it's negative than the interval is decreasing. |
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How do you find the inflection points and concavity of a function?
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1. Find where f '' (x) = 0.
2. Create a number line and either plug in points or use trends to find out if the space between the points are positive or negative. 3. If the gap is positive, then the interval is concave up. If the gap is negative, then the interval is concave down. 4. To find the points of inflection, simply plug in the x-values that you solved into the original equation. |
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How would you proceed in the process of optimization?
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1. Understand the problem
2. Make a sketch 3. Introduce notation 4. Write a function for quantity to be optimized. 5. Write the function in terms of one variable and consider the interval. 6. Find the absolute max or min by setting the equation f ' (x) = 0. |
