Chapter 5: Applications of the Derivative Chapter 4: Derivatives Chapter 5: Applications
Objectives: To be able to use the derivative to analyze function Draw the graph of the function based on the analysis Apply the principles learned to problem situations
Example 1: Analyze and sketch y = x 5 - 5x
Example 2: Make the curve y = ax 4 + bx 3 + cx 2 + dx + e pass through the points (0,3) and (-2,7) and at (-1,4) have a point of inflection with a horizontal tangent line.
Example 3: Analyze and sketch a. y = 3x 5 - 5x 3 b. y = x 4 + 2x c. y = 2x 3 - 3x x + 9
Example 4: On what interval is the function y = x 3 -4x 2 +5x concaved upward?
Example 5: Find the points of the curve, y = x 3 - x 2 – x +1, where the tangent line is horizontal.
Example 6: Find a cubic equation y = ax 3 +bx 2 +cx+d whose graph has a horizontal tangent line at (-2, 6) and (2, 0) and also find the point of inflection.
Example 7: Find the equations of both lines through the point (2, -3) that are tangent to the parabola y = x 2 + x.
Example 8: Make the curve y = ax 3 + bx 2 + cx+d pass through (0, 3) and have at (1, 2) a point of inflection with a horizontal tangent.
Example 9: Make the curve y = ax 4 + bx 3 + cx 2 + dx + e have a critical point at (0, 3) and have an inflection point at (1, 2) with inflection tangent 6x + y = 8.
Example 10: Make the curve y = ax 3 + bx 2 + cx + d pass through (-1, -1) and have at (1, 3) an inflection point with inflectional tangent 4x – y = 1.
Example 11: Find a, b, c and d such that the function defined by y = ax 3 + bx 2 + cx + d will have a relative extrema at (1, 2) and (2, 3).
Example 12: If y = ax 3 + bx 2 + cx + d determine a, b, c and d so that the function will have a relative extremum at (0, 3) and so that the graph of the function will have a point of inflection at (1, -1).
Example 13: Find the parabola with equation y = ax 2 + bx whose tangent line at (1, 1) has equation y = 3x - 2.
Example 14: Find a cubic equation, y = ax 3 + bx 2 + cx + d whose graph has horizontal tangents at (-2, 6) and (2, 0).