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Parametric Equations t x y
Eliminating the Parameter 1) 2)
11.2 Slope and Concavity For the curve given by Find the slope and concavity at the point (2,3) At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up
Horizontal and Vertical tangents A horizontal tangent occurs when dy/dt = 0 but dx/dt 0. A vertical tangent occurs when dx/dt = 0 but dy/dt 0. Vertical tangents Horizontal tangent
Polar Coordinate Plane
Figure Pole Polar axis Polar Coordinates
Polar/Rectangular Equivalences x 2 + y 2 = r 2 tan θ = y/x x = r cos θ y = r sin θ θ)
Figure 9.40(a-c). Symmetries
Figure 9.42(a-b). Graph r 2 = 4 cos θ
Figure Finding points of intersection Third point does not show up. On r = 1-2 cos θ, point is (-1, 0) On r = 1, point is (1, π)
Slope of a polar curve Where x = r cos θ = f(θ) cos θ And y = r sin θ = f(θ) sin θ Horizontal tangent where dy/dθ = 0 and dx/dθ≠0 Vertical tangent where dx/dθ = 0 and dy/dθ≠0
Finding slopes and horizontal and vertical tangent lines For r = 1 – cos θ (a) Find the slope at θ = π/6 (b) Find horizontal tangents (c) Find vertical tangents
r = 1 – cos θ
Find Horizontal Tangents
Find Vertical Tangents Horizontal tangents at: Vertical tangents at:
Figure Finding Tangent Lines at the pole r = 2 sin 3θ r = 2 sin 3θ = 0 3θ = 0, π, 2 π, 3 π θ = 0, π/3, 2 π/3, π
Figure Area in the Plane
Figure Area of region
Figure Find Area of region inside smaller loop
Figure Area between curves
Length of a Curve in Polar Coordinates Find the length of the arc for r = 2 – 2cosθ sin 2 A =(1-cos2A)/2 2 sin 2 A =1-cos2A 2 sin 2 (1/2θ) =1-cosθ
THEOREM 2 Sum of a Geometric Series Let c 0. If |r| < 1, then If |r| ≥ 1, then the geometric series diverges. Sum of an Infinite Geometric Series (80)
Y – Intercept of a Line The y – intercept of a line is the point where the line intersects or “cuts through” the y – axis.
The Calculus of Parametric Equations As we have done, we will do again. Parametric curves have tangent lines, rates of change, area under and above, local.
Horizontal Lines Vertical Lines Lines, Lines, Lines!!! ~
When you see… Find the zeros You think…. To find the zeros...
3.9 Derivatives of Exponential and Logarithmic Functions.
1. If f(x) =, then f (x) = 2. Advanced Placement Calculus Semester One ReviewName:___________________ Calculator Active Multiple Choice 3. If x 3 + 3xy.
3.7 Equations of Lines in the Coordinate Plane The slope m of a line is the ratio `of the vertical change (rise) to the horizontal change (run) between.
Warm Up No Calculator 2) A curve is described by the parametric equations x = t 2 + 2t, y = t 3 + t 2. An equation of the line tangent to the curve at.
UNIT 2 LESSON 8 CHAIN & QUOTIENT RULE 1. 2 Chain Rule and Quotient Rule Example 1 dy = (2x 2 – 1) 2 [4(1 + 3x) 3 (3)] – (1 + 3x) 4 [2(2x 2 – 1)(4x)] dx.
A(a,b) y = mx +c y = f(x) EQUATIONS OF TANGENTS tangent NB: at A(a, b) gradient of line = gradient of curve gradient of line = m (from y = mx + c ) gradient.
3.7 Implicit Differentiation Implicitly Defined Functions –How do we find the slope when we cannot conveniently solve the equation to find the functions?
Unit 6 – Fundamentals of Calculus Section 6.4 – The Slope of a Curve No Calculator.
MATHPOWER TM 12, WESTERN EDITION Chapter 3 Conics.
Disks, Washers, and Cross Sections Review. Let R be the region in the first quadrant under the graph of c)Setup but do not evaluate the integral necessary.
DERIVATIVE OF ARC LENGTH Let y = f(x) be the equation of a given curve. Let A be some fixed point on the curve and P(x,y) and Q(x+Δx, y+ Δy) be two neighbouring.
2-3 Slope Slope 4 indicates the steepness of a line. 4 is “the change in y over the change in x” (vertical over horizontal). 4 is the ‘m’ in y = mx + b.
Circles Graphing and Writing Equations What is a circle? A conic formed when …. A second degree equation… A locus of points…
EXAMPLE 1 Write an equation of a line from a graph SOLUTION m 4 – (– 2) 0 – 3 = 6 – 3 = = – 2 STEP 2 Find the y -intercept. The line intersects the y -axis.
E-learning extended learning for chapter 11 (graphs)
Finding the Slope of a Line From Two Points. Given the points: (3, 4) and (2, 6) Find the slope of the line. y 2 – y 1 x 2 – x 1 = x 1, y 1 x 2, y 2 6.
Polar Coordinates We Live on a Sphere. Polar Coordinates Up till now, we have graphed on the Cartesian plane using rectangular coordinates In the rectangular.
What is it?. Definition: A hyperbola is the set of points P(x,y) in a plane such that the absolute value of the difference between the distances from.
Problems on tangents, velocity, derivatives, and differentiation.
DO NOW: Find where the function f(x) = 3x 4 – 4x 3 – 12x is increasing and decreasing.
Warm Up A particle moves vertically(in inches)along the x-axis according to the position equation x(t) = t 4 – 18t 2 + 7t – 4, where t represents seconds.
Graphing Data on the Coordinate Plane 1-9. Two number lines that intersect at right angles form a coordinate plane. The horizontal axis is the x-axis.
Arc Length Cartesian, Parametric, and Polar. Arc Length x k-1 xkxk Green line = If we do this over and over from every x k—1 to any x k, we get.
(r, ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.
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