Section4-3

# Section4-3 - Section 4.3 Derivatives and the Shape of...

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1 Section 4.3 Derivatives and the Shape of Curves • Goals – Apply the Mean Value Theorem to finding where functions are increasing and decreasing – Discuss the • first derivative test and • second derivative test for local max/minima Mean Value Theorem • This theorem is the key to connecting derivatives with the shape of curves: Geometric (cont’d) Geometric (cont’d) • Thus Equation 1 of the Theorem says that there is at least one point P ( c , f ( c )) on the graph where – the slope of the tangent line is the same as – the slope of the secant line AB . • In other words, there is a point P where the tangent line is parallel to the secant line AB . Example • If an object moves in a straight line with position function s = f ( t ) , then the average velocity between t = a and t = b is and the velocity at t = c is f ( c ) . • Thus the Mean Value Theorem says that… ( ) () fb fa ba Example (cont’d) • …at some time t = c between a and b the – instantaneous velocity f ( c ) is equal to the – average velocity. • For instance, if a car traveled 180 km in 2 h, then the speedometer must have read 90 km/h at least once .

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2 Increasing/Decreasing Functions • Earlier we observed from graphs that a function with a positive derivative is increasing.
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Section4-3 - Section 4.3 Derivatives and the Shape of...

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