Chapter 9 Math

Find min and max values of f.

Ex) f(x) = 2sinx - cos2x f'(x) = 2 d/dx sinx - d/dx cos2x = 2cosx -(-sin2x) d/dx 2x = 2cosx + 2sin2x = 2cosx + 4sinxcosx 0 = 2cosx(1 + 2sinx) Cosx = 0 or sinx = -1/2 x = 90,270 or x=210,330 Plug in these degrees to the original function and show the max and min

When graphing in a polar coordinate system

Ex) graph (3,180 degrees) find 180 degrees and count 3 lines Ex) graph (3, -30 degrees) find 330 degrees and count 3 lines Ex) graph (-3,120 degrees) find 120 degrees, count 3 lines, and reflect over the x-axis(look across the circle)

Convert polar equations to rectangular form

Ex) theta = pi/3 tan theta = y/x tan pi/3 = root 3 Y/x = root 3 Y = x root 3 Ex) r = 4sin theta r = 4 • y/r r^2 = 4y x^2 + y^2 = 4y move to one side x^2 + (y-2)^2 = 4

Find dy/dx by using substitution of u

Ex) y = cosx^2 Let x^2 = u du/dx = 2x y = cosu y' = -sinu d/dx u y' = -sinx^2 • 2x y'=-2xsinx^2

Prove the corollary to theorem 4 by giving specific reasons why each of the following functions is continuous at each number in its domain. (Tangent)

Let c be any real number in the domain of tangent function. Then lim x->c tanx= lim sinx/cosx = sinc/cosc = tanc Since sine and cosine are continuous everywhere, and since lim x->c tanx = tanc, the tangent function is continuous at every member of its domain.

Ex of lim

Lim x->0(3x+1-cosx)/4x lim(3x/4x) + 1/4(lim 1-cosx/x) 3/4 + 1/4(0) 3/4

Graph each polar equation

Make a chart with 3 columns of theta, the part with sin/cos (Ex:r=1+2sintheta so put sintheta in the center column), put the original function in the last column. Do all theta of 0 degrees to 90 degrees in factors of 15 degrees. The second quadrant should be the reflection of the first quadrant. Do the degrees of factors of 15 for the 3rd quadrant in the table as well. ***Test out degrees from all quadrants just to be sure they are symmetrical or not. If it is cos theta it will be that the 1st and 4th quadrant are symmetrical because cos theta in the second quadrant is negative so would reflect to the 4th quadrant. SIN THETA IS SYMMETRICAL WITH 1st/2nd QUADRANT

Graph of a polar equation is symmetric with respect to the polar axis

Replace (r,theta) by (r,-theta) or (-r,pi-theta) Graph looks like a reflection over the x-axis with 1st/4th quadrant

Graph of a polar equation is symmetric with respect to the pole

Replace (r,theta) by (r,pi+theta) or (-r,theta) Graph looks like a 180 degree rotation with 1st/3rd quadrant

Graph of a polar equation is symmetric with respect to line theta=pi/2

Replace (r,theta) by (r,pi-theta) or (-r,-theta) The graph looks like a reflection over the y-axis with 1st/2nd quadrant

Theorem 4

The sine and cosine functions are continuous everywhere.

Derivatives of trigonometric functions

d/dx sinx = cos x d/dx cosx = -sinx d/dx tanx = sec^2x d/dx cotx = -csc^2x d/dx secx = secxtanx d/dx cscx = -cscxcotx

Sketch the curve defined by x=root t; y=roots 1+t

define the domain/range x>=0 (b/c square root); y>=0 substitute into one equation. x=root t. y=root 1+t x^2 = t. y^2 = 1+t. Substitute t y^2 = 1 + x^2 y^2-x^2 = 1. Graph the hyperbola within the domain/range

Rates and extrema problems

dx/dt = rate dx/dt = dx/dtheta x dtheta/dt

When it says use the definition of derivative to prove: d/dx cosx = -sinx

f(x) = lim h->0 (f(x+h)-f(x)/h) Let f(x) = cosx; f(x+h) = cos(x+h) f(x) = lim h->0 (cos(x+h) - cosx)/h = lim h->0 (cosxcosh-sinxsinh-cosx)/h = lim h->0 (-cosx(1-cosh)/h - sinxsinh/h) = (-1)cosx x 0 - sinx x 1 = -sinx

From rectangular to polar formulas

r = +- square root of x^2 + y^2 cos theta = x/r sin theta = y/r

Parametric equations

x = f(t) y = g(t)

From polar to rectangular formulas

x=rcos theta y=rsin theta

Polar form

z = r(cos theta + isin theta)

Identify the curve C defined by x=cos t and y=sec t by eliminating the parameter t. Then draw C.

-1<=x<=1. -1<=y<=1 x= cos t. y=1/cos t y=1/x xy= 1 Make a data table and plot points. Define the graph as a hyperbola.

Theorem 5

1. Lim x->0 sinx/x = 1 2. Lim x->0 (1-cosx)/x = 0

Polar coordinates

(x,y) -> (r,theta)

Ex of rates and extrema problem

A lighthouse is located on a small island 3km away from the nearest point P on a straight shoreline and it's light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1km from P? (Triangle: x is the base, 3km is the height, and theta is at the top corner) dtheta/dt = 4rev/min • 2pirad/rev = 8pi rad/min tan theta = x/3. X=3tantheta dx/dt = 3sec^2theta • 8pi (dx/dtheta • dtheta/dt) dx/dt = 3(root 10/3)^2 • 8pi (you get the root 10/3 by finding sec theta) dx/dt = 83.7 km/min

Convert rectangular equation to polar form

Ex) y= root 3(x) divide both sides by x y/x = root 3. (x,y) = (cos theta, sin theta) tan theta = root 3 theta = pi/3 + pi • k ; k is an integer

Find a pair of parametric equations to describe the position P of a particle in a circle with radius r and center (1,1)

x^2 + y^2 = r^2 x-1 = rcos theta and y-1 =rsin theta (x-1)^2 + (y-1)^2 = r^2

Quotient rule for derivative

y' = (denominator(d/dx numerator) - d/dx den(num))/(den)^2