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: Find the equation of the tangent

line to the parabola, y = x1/2 at a. (0,0) b. (1,1).

Example 2: Find the point on the parabola

y = x2 – 2x + 1 where the tangent line is horizontal.

Example 3: Locate the point where the

tangent line is a. horizontal b. Vertical.

x3 + y3 = 6xy

Example 4: Find the equation of the tangent

line to the ellipse at the end of the latus rectum found in the first quadrant. Equation of ellipse is x2/16 + y2/25 = 1

Example 5: Find the equation of the tangent

line at t = 0. x = t2 + t y = t2 - t

Example 6: Find the equation of the tangent

line to y = 1 + x – 2x3 at x = 1.

Example 7: Find the equation of the tangent

line to y = x / (2x – 1) at x = 1.

Example 8: Find the equation of the tangent

line to y = 2 /(3-x)1/2 at x = - 1.

Example 9: Find the equation of the tangent

line to y = x / (x2-3) at x = 2.

Example 10: Find the equation of the

horizontal tangent line of x = t(t2 – 3) y = 3(t2 – 3)

Example 11: 2(x2+y2)2 = 25(x2 –y2) is an

equation of a curve called the lemniscate.

(a) find the equation of its tangent line at (3, 1).

(b) Locate the points where the tangent line is horizontal.

Example 12: Find the equation of the tangent

line at the given point. x2/16 - y2/9 = 1 at ( -

5 , 9/4 )

Example 13: Find the equation of the tangent

line at the given point.y2 = 5x4 – x2 at ( 1 , 2

)

Example 14: Find the equation of both lines

through ( 2, - 3) that are both tangent to the parabola y = x2 + x.

Example 15: Where does the normal line to the

parabola y = x – x2 at ( 1 , 0 ) intersect the curve a second time?

Example 16: Find the cubic function y = ax3 +

bx2 + cx + d where its graph has horizontal lines ( -2 , 6) and ( 2, 0).

Example 17: The vertex of a parabola is the

point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of y2 = -2x + 8

Example 18: The vertex of a parabola is the

point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of y = x2 + 8x – 5.

Example 19: The vertex of a parabola is the

point where the tangent line is either horizontal or vertical (axis is not oblique). Locate the vertex of x2

– 4x + y = 0.

Example 20: Locate the points where the

tangent line is horizontal. y = x3 – x2 – x + 1

Example 20: Locate the points where the

tangent line is horizontal. y = 2x3 – 3x2 – 6x + 37