Chapter 3.7: Derivatives of the Trigonometric Functions

Chapter 3.7: Derivatives of the Trigonometric Functions

Expected Skills:

? Know (and be able to derive) the derivatives of the 6 elementary trigonometric functions.

? Be able to use the product, quotient, and chain rules (where appropriate) to differentiate functions involving trigonometry.

? Be able to use the derivative to calculate the instantaneous rates of change of a trigonometric function at a given point.

? Be able to use the derivative to calculate the slope of the tangent line to the graph of

a trigonometric function at a given point.

? Be able to use the derivative to calculate to answer other application questions, such

as local max/min or absolute max/min problems.

Practice Problems:

1. Fill in the given table:

f (x)

sin x

cos x

tan x

cot x

sec x

csc x

f (x)

f 0 (x)

sin x

cos x

cos x

? sin x

tan x

sec2 x

cot x

? csc2 x

sec x sec x tan x

csc x ? csc x cot x

1

f 0 (x)

2. Use the definition of the derivative to show that

d

(cos x) = ? sin x

dx

d

cos (x + h) ? cos x

(cos x) = lim

h¡ú0

dx

h

cos x cos h ? sin x sin h ? cos x

= lim

h¡ú0

h





cos x cos h ? cos x sin x sin h

= lim

?

h¡ú0

h

h





cos h ? 1

sin h

? sin x

= lim cos x

h¡ú0

h

h

= (cos x)(0) ? (sin x)(1)

= ? sin x

3. Use the quotient rule to show that

d

(cot x) = ? csc2 x.

dx

d

d  cos x 

(cot x) =

dx

dx sin x

(sin x)(? sin x) ? (cos x)(cos x)

=

sin2 x

2

?(sin x + cos2 x)

=

sin2 x

1

=? 2

sin x

= ? csc2 x

4. Use the quotient rule to show that

d

(csc x) = ? csc x cot x.

dx





d

d

1

(csc x) =

dx

dx sin x

(sin x)(0) ? (1)(cos x)

=

sin2 x

cos x

=? 2

sin x

1 cos x

=?

sin x sin x

= ? csc x cot x

2







+ h ? tan ¦Ð3

by interpreting the limit as the derivative of a

5. Evaluate lim

h¡ú0

h

function at a particular point.







¦Ð 

tan ¦Ð3 + h ? tan ¦Ð3

d

lim

=

(tan x)

= sec2

=4

h¡ú0

h

dx

3

x= ¦Ð

tan

¦Ð

3

3

For problems 6-16, differentiate

6. f (x) = 2 cos x + 4 sin x

?2 sin x + 4 cos x

sin2 x

7. f (x) =

cos x

2 sin x + sin x tan2 x

8. f (x) = x3 sin x

3x2 sin x + x3 cos x

9. f (x) = sec2 x + tan2 x

4 sec2 (x) tan (x)

 

1

10. f (x) = tan

x2

 

1

?3

2

?2x sec

x2

11. f (x) = sec 2x

2 sec (2x) tan (2x)

12. f (x) = cos3 3x

?9 sin (3x) cos2 (3x)

¦Ð 

13. f (x) = sin

x

¦Ð 

?¦Ðx?2 cos

x

14. f (x) = sin (sin 2x)

2 cos (sin 2x) cos 2x

3

15. f (x) = tan2 (x2 ? 1)

4x tan (x2 ? 1) sec2 (x2 ? 1)

16. f (x) = 4x2 csc 5x

8x csc (5x) ? 20x2 csc (5x) cot (5x)

 ¦Ð i

d h ¡Ì

g

x

17. Use the following table to calculate

2 sin

dx

4

x f (x) f 0 (x)

1 ?2

?5

5

?3

2

3 ?1

6

3

1

4

5

4

7

?

x=3

g(x) g 0 (x)

3

9

4

?2

7

?6

?2

5

1

8

9¦Ð

4

18. What is the 100th derivative of y = sin (2x)?

2100 sin 2x

19. Compute an equation of the line which is tangent to the graph of f (x) =

point where x = ¦Ð.

y=

cos x

at the

x

2

1

x?

2

¦Ð

¦Ð

20. Find all points on the graph of y = sin2 x where the tangent lines are parallel to the

line y = x.

¦Ð

+ ¦Ðk where k is any integer

4

For problems 21-22, find all values of x in the interval [0, 2¦Ð] where the graph of

the given function has horizontal tangent lines.

21. f (x) = sin x cos x

¦Ð 3¦Ð 5¦Ð 7¦Ð

, , ,

4 4 4 4

22. g(x) = csc x

¦Ð 3¦Ð

,

2 2

4

23. Use the Intermediate Value Theorem to show that there is at least one point in the

1

interval (0, 1) where the graph of f (x) = sin x ? x3 will have a horizontal tangent

3

with plots

line.

animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d,

(1)

conformal3d, contourplot, contourplot3d,

coordplot3d, densityplot,

f 0 (x) = cos x ? x2 . conformal,

Firstly,

notice that f 0 (x) is coordplot,

continuous

for all x; therefore, it

display, dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d,

0

is continuous for all inequal,

x ininteractive,

[0, 1]. interactiveparams,

Secondly,intersectplot,

notice listcontplot,

that flistcontplot3d,

(0) = 1 > 0 and f 0 (1) =

listdensityplot, listplot, listplot3d, loglogplot, logplot, matrixplot, multiple, odeplot, pareto,

cos (1) ? 1 < 0. Thus,

the Intermediate Value Theorem states there is at least one x0

plotcompare, pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d,

in the interval (0, 1) polyhedra_supported,

with f 0 (x0 ) =polyhedraplot,

0. In other

at least one x0 in (0, 1)

rootlocus,words,

semilogplot,there

setcolors,is

setoptions,

setoptions3d, spacecurve, sparsematrixplot, surfdata, textplot, textplot3d, tubeplot

where f (x) will have a horizontal tangent

line.

Pi

a d plot sqrt 2 sin x , x = 0 ..

¡Ì

2

, scaling = constrained

PLOT ...

24. Considerh the igraphs of f (x) = 2 cos(x)

and

g(x) =

Pi

b

d

plot

sqrt

2

cos

x

,

x

=

0

..

,

scaling

= constrained

¦Ð

2

interval 0, .

PLOT ...

2

display a, b

¡Ì

(2)

2 sin(x) shown below

on the

(3)

1.4

1.2

1

0.8

0.6

0.4

0.2

0

¦Ð

16

¦Ð

8

3¦Ð

16

¦Ð

4

x

5¦Ð

16

3¦Ð

8

7¦Ð

16

¦Ð

2

¦Ð

Show that the graphs of f (x) and g(x) intersect at a right angle when x = . (Hint:

4

¦Ð

Show that the tangent lines to f and g at x = are perpendicular to each other.)

4

¦Ð 

¦Ð 

¦Ð

f0

= ?1 and g 0

= 1. So, the tangent lines to f and g at x =

are

4

4

4

perpendicular to one another since the product of their slopes is ?1.

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download