Derivatives Lesson Oct 14
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1. The document discusses distance-time and velocity-time graphs and their relationships. It defines acceleration as the derivative of velocity over time. 2. It provides an example distance-time function and asks to sketch the corresponding velocity-time graph. It also discusses how to determine if a function is increasing or decreasing based on the signs of its derivative. 3. The document ends by giving a velocity function and asking to graph its derivative and determine where the original function is increasing, where it is concave up, and to sketch a possible graph based on an initial value.
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- 2. Given the following distance time graph represented by the function: 1 3 f (x) = x + 1 x 2 3 8 distance time Sketch a graph for the velocity time graph.
- 3. Velocity time graph comes from the derivative of the distance time graph. 2 1 f '(x) = x + 4 x velocity time
- 4. Acceleration describes how fast the velocity is changing with time. acceleration time
- 5. Since f '(x) is a function we can calculate it's derivative: f '(a + h) f '(a) f ''(a) = lim h 0 = (f ') ' (a) h f ''(a) is called the second derivative of f at a also written as: d (dy) d2y = dx dx dx2
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- 7. Remember that the derivative of a function tells you whether the function is increasing or decreasing. Since f '' is the derivative of f ' 1. If f ''(x) > 0 on an interval, then f ' is increasing 2. If f ''(x) < 0 on an interval, then f ' is decreasing
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