TS: Explicitly assessing information and drawing conclusions Increasing & Decreasing Functions

Objectives To examine the relationship between the slope of tangent lines and the behavior of a curve. To examine the relationship between the slope of tangent lines and the behavior of a curve. To determine when a function is increasing, decreasing, or neither. To determine when a function is increasing, decreasing, or neither. To find the critical points of a function. To find the critical points of a function. To determine the intervals on which a function is increasing or decreasing. To determine the intervals on which a function is increasing or decreasing.

The Derivative The derivative is used to find: Instantaneous Rate of Change Instantaneous Rate of Change Slopes of Tangent Lines Slopes of Tangent Lines

Tangent Lines The line tangent to the curve of a function emulates the behavior of the curve near the point of tangency. The line tangent to the curve of a function emulates the behavior of the curve near the point of tangency.

Tangent Lines

Behavior of a Curve Always “read” the graph from left to right.

Behavior of a Curve The curve increases until it reaches a summit.

Behavior of a Curve The curve decreases until it reaches a valley.

Behavior of a Curve The curve increases again.

Behavior of a Curve Question: How can you determine where the curve is increasing or decreasing? Question: How can you determine where the curve is increasing or decreasing? Answer: Study the tangent lines. Answer: Study the tangent lines. On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing. On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing.

Tangent Lines Tangent line is positively sloped – function is increasing.

Tangent Lines Tangent line is positively sloped – function is still increasing.

Tangent Lines Tangent line levels off at the summit.

Tangent Lines Tangent line is negatively sloped – function is decreasing.

Tangent Lines Tangent line is negatively sloped – function is still decreasing.

Tangent Lines Tangent line levels off at the valley..

Tangent Lines Tangent line is positively sloped – function is increasing again.

Positive Derivative  Function Increasing

Negative Derivative  Function Decreasing

The Derivative If f’ (x) > 0, then f (x) is increasing. If f’ (x) > 0, then f (x) is increasing. if f’ (x) < 0, then f (x) is decreasing. if f’ (x) < 0, then f (x) is decreasing.

Behavior of a Curve Question: What if the derivative equals 0? Question: What if the derivative equals 0? Answer: The function is neither increasing nor decreasing. Answer: The function is neither increasing nor decreasing. Values that make the derivative of a function equal zero are candidates for the location of maxima and minima of the function. Values that make the derivative of a function equal zero are candidates for the location of maxima and minima of the function.

Behavior of a Curve Tangent line has a slope of 0 at the summit.

Behavior of a Curve Tangent line has a slope of 0 at the valley.

Max & Min

Behavior of a Curve Consider: Consider: What is the function doing at x = 0 and at x = 10 ? What is the function doing at x = 0 and at x = 10 ? The function is decreasing through x = 0.

Behavior of a Curve The function is increasing through x = 10.

Critical Points cusp point The derivative is not defined. Neither a max nor a min. x1x1 x2x2 x3x3 x5x5 x4x4 x y

Critical Points y x1x1 x2x2 x3x3 x5x5 x4x4 x

Critical points are the places on a function where the derivative equals zero or is undefined. Critical points are the places on a function where the derivative equals zero or is undefined. Interesting things happen at critical points. Interesting things happen at critical points.

Critical Points Steps to find critical points: Steps to find critical points: 1. Take the derivative. 2. Set the derivative equal to zero and solve. 3. Find values where the derivative is undefined. Set the denominator of the derivative equal to zero to find points where the derivative could be undefined. Set the denominator of the derivative equal to zero to find points where the derivative could be undefined.

Critical Points Find the critical points of: Find the critical points of:

Critical Points Find the critical points of: Find the critical points of:

Critical Points Find the critical points of: Find the critical points of:

Increasing & Decreasing Find the intervals on which the function is increasing or decreasing: Find the intervals on which the function is increasing or decreasing:

Increasing & Decreasing 0 Decreasing: Increasing:

Increasing & Decreasing Find the intervals on which the function is increasing or decreasing: Find the intervals on which the function is increasing or decreasing:

Increasing & Decreasing 0 0 Decreasing: Increasing:

Increasing & Decreasing Find the intervals on which the function is increasing or decreasing: Find the intervals on which the function is increasing or decreasing:

Increasing & Decreasing UND. Decreasing: Increasing:

Conclusion The derivative is used to find the slope of the tangent line. The derivative is used to find the slope of the tangent line. The line tangent to the curve of a function emulates the behavior of the curve near the point of tangency. The line tangent to the curve of a function emulates the behavior of the curve near the point of tangency. On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing. On an interval, the sign of the derivative of a function indicates whether that function is increasing or decreasing.

Conclusion f (x) is increasing if f’ (x) > 0. f (x) is increasing if f’ (x) > 0. f (x) is decreasing if f’ (x) < 0. f (x) is decreasing if f’ (x) < 0. Values that make the derivative of a function equal zero are candidates for the location of maxima and minima of the function. Values that make the derivative of a function equal zero are candidates for the location of maxima and minima of the function.

Conclusion Critical points are the places on a graph where the derivative equals zero or is undefined. Critical points are the places on a graph where the derivative equals zero or is undefined. First derivative  Positive  Increasing First derivative  Positive  Increasing First derivative  Negative  Decreasing First derivative  Negative  Decreasing