b14 primary trig derivatives

8/11/2019 B14 Primary Trig Derivatives

http://slidepdf.com/reader/full/b14-primary-trig-derivatives 1/21

B1.4 - Derivatives of Primary Trigonometric Functions

IB Math HL/SL - Santowski



(A) Derivative of the Sine Function -Graphically

We will predict the what the derivativefunction of f(x) = sin(x) looks like from ourcurve sketching ideas:We will simply sketch 2 cycles(i) we see a maximum at /2 and -3 /2

derivative must have x-intercepts(ii) we see intervals of increase on (-2 ,-

3 /2), (- /2, /2), (3 /2,2 ) derivativemust increase on this intervals(iii) the opposite is true of intervals ofdecrease(iv) intervals of concave up are (- ,0) and( ,2 ) so derivative must increase onthese domains(v) the opposite is true for intervals of

concave up

So the derivative function must look like the cosine function!!



(B) Derivative of Sine Function - Algebraically

We will go back to our limit concepts for determining the derivative of y = sin(x)algebraically

hh

xhh

x xdxd

xh

hhh

x xdxd

h xh

hh x

xdxd

h xhh x

xdxd

h x xhh x

xdxd

h xh x

xdxd

h x f h x f

x f

hh

hhhh

hh

h

h

h

h

)sin(lim)cos(

1)cos(lim)sin()sin(

)cos(lim)sin(

lim1)cos(

lim))(sin(lim)sin(

)cos()sin(lim

]1))[cos(sin(lim)sin(

)cos()sin()]1))[cos(sin(lim)sin(

)sin()cos()sin()cos()sin(lim)sin(

)sin()sin(lim)sin(

)()(lim)(

00

0000

00

0

0

0

0



(B) Derivative of Sine Function - Algebraically

So we come across 2 special trigonometric limits:

and

So what do these limits equal?

We will introduce a new theorem called a Squeeze (orsandwich) theorem if we that our limit in question liesbetween two known values, then we can somehow “squeeze”the value of the limit by adjusting/manipulating our two knownvalues

So our known values will be areas of sectors and triangles sector DCB, triangle ACB, and sector ACB

h

h

h

)sin(lim

0 h

h

h

1)cos(lim

0



(C) Applying “Squeeze Theorem” to Trig.Limits

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

1.5 -1 -0.5 0.5 1 1.5

D

B = (cos(x), 0)CE = (1,0)

A = (cos(x), sin(x))




We have sector DCB and sector ACB “squeezing” the triangle ACBSo the area of the triangle ACB should be “squeezed between”the area of the two sectors

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.5 -1 -0.5 0.5 1 1.5

D

B = (cos(x), 0)C

E = (1,0)

A = (cos(x), sin(x))




Working with our area relationships (make h = )

We can “squeeze or sandwich” our ratio of sin(h) / h between cos(h)and 1/cos(h)

)cos(1)sin()cos(

)cos()cos()cos()sin(

)cos()(cos

)cos()sin()(cos

)1(2

1)cos()sin(2

1)(cos2

1

)()(21))((2

1)()(21

2

2

22

22

OC OAOBOB




Now, let s apply the squeeze theorem as we take our limits as h0+ (and since sin(h) has even symmetry, the LHL as h 0 - )

Follow the link to Visual Calculus - Trig Limits of sin(h)/h to seetheir development of this fundamental trig limit

1)sin(

lim

1)sin(

lim1

)cos(

1lim

)sin(lim)cos(lim

0

0

000

h

hh

hhh

hh

h

h

hhh

http://archives.math.utk.edu/visual.calculus/2/trig.1/1.html








Now what about (cos(h) – 1) / h and its limit we will treat this algebraically

011

011

1)cos()sin(

lim)sin(

lim1

1)cos()(sin

lim

1)cos(1)(cos

lim

1)cos(

1)cos(1)cos(lim

1)cos(lim

00

2

0

2

0

0

0

h

h

h

h

hh

h

hh

hhh

hh

h

h

hh

h

h

h

h



(D) Fundamental Trig. Limits Graphicand Numeric Verification

x y -0.05000 0.99958 -0.04167 0.99971 -0.03333 0.99981 -0.02500 0.99990 -0.01667 0.99995 -0.00833 0.99999 0.00000 undefined 0.00833 0.99999 0.01667 0.99995

0.02500 0.99990 0.03333 0.99981 0.04167 0.99971 0.05000 0.99958



(D) Derivative of Sine Function

Since we have our two fundamental trig limits, we can now go backand algebraically verify our graphic “estimate” of the derivative of thesine function:

)cos()sin(

1)cos(0)sin()sin(

)sin(lim)cos(

1)cos(lim)sin()sin(

01)cos(

lim

1)sin(

lim

00

0

0

x xdxd

x x xdxd

hh

xhh

x xdxd

hhh

h

hh

h

h



(E) Derivative of the Cosine Function

Knowing the derivative of the sine function,we can develop the formula for the cosinefunction

First, consider the graphic approach as wedid previously




We will predict the what the derivativefunction of f(x) = cos(x) looks like fromour curve sketching ideas:We will simply sketch 2 cycles(i) we see a maximum at 0, -2 & 2 derivative must have x-intercepts(ii) we see intervals of increase on (- ,0),

( , 2 ) derivative must increase onthis intervals(iii) the opposite is true of intervals ofdecrease(iv) intervals of concave up are (-3 /2,-

/2) and ( /2 ,3 /2) so derivative mustincrease on these domains(v) the opposite is true for intervals of

concave upSo the derivative function must look like

some variation of the sine function!!




We will predict the what the derivativefunction of f(x) = cos(x) looks like fromour curve sketching ideas:We will simply sketch 2 cycles(i) we see a maximum at 0, -2 & 2 derivative must have x-intercepts(ii) we see intervals of increase on (- ,0),

( , 2 ) derivative must increase onthis intervals(iii) the opposite is true of intervals ofdecrease(iv) intervals of concave up are (-3 /2,-

/2) and ( /2 ,3 /2) so derivative mustincrease on these domains(v) the opposite is true for intervals of

concave upSo the derivative function must look like

the negative sine function!!




Let s set it up algebraically:

)sin(1)sin()cos(

)1(2

cos)cos(

22sin

2

)cos(

2sin)cos(

x x xdxd

x xdx

d

xdxd x

xd

d xdxd

xdxd

xdxd



(F) Derivative of the Tangent Function -Graphically

So we will go through ourcurve analysis againf(x) is constantly increasingwithin its domain

f(x) has no max/min pointsf(x) changes concavity fromcon down to con up at 0,+ f(x) has asymptotes at +3

/2, + /2



(F) Derivative of the Tangent Function -Graphically

So we will go through ourcurve analysis again:F(x) is constantly increasingwithin its domain f `(x)should be positive within itsdomainF(x) has no max/min points f „(x) should not have roots F(x) changes concavity fromcon down to con up at 0,+ f „(x) changes from decrease toincrease and will have a minF(x) has asymptotes at +3 /2, + /2 derivative shouldhave asymptotes at the samepoints



(F) Derivative of the Tangent Function - Algebraically

We will use the fact that tan(x) = sin(x)/cos(x) to find the derivative of tan(x)

x x

xdxd

x x x xdxd

x x x x x

xdxd

x x xdx

d x xdx

d

xdxd

x x

dxd

xdxd

22

2

22

2

2

seccos

1)tan(

cos sincos)tan(

cos)sin()sin()cos()cos(

)tan(

)cos()sin()cos()cos()sin()tan(

)cos()sin(

)tan(



(H) Homework

Stewart, 1989, Chap 7.2, Q1-5,11

b14 primary trig derivatives

Documents