Chapter 2.5 Practice Problems

Chapter 2.5 Practice Problems

EXPECTED SKILLS: ? Know the derivatives of the 6 elementary trigonometric functions. ? Be able to use these derivatives in the context of word 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 (x) sin x cos x tan x cot x sec x csc x

f (x) cos x - sin x sec2 x - csc2 x sec x tan x - csc x cot x

d 2. Use the definition of the derivative to show that (cos x) = - sin x

dx Hint: cos ( + ) = cos cos - sin sin

d

cos (x + h) - cos x

(cos x) = lim

dx

h0

h

cos x cos h - sin x sin h - cos x

= lim

h0

h

cos x cos h - cos x sin x sin h

= lim

-

h0

h

h

cos h - 1

sin h

= lim cos x

- sin x

h0

h

h

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

= - sin x

1

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

-(sin2 x + cos2 x)

=

sin2 x

1 = - sin2 x

= - csc2 x

d 4. Use the quotient rule to show that (csc x) = - csc x cot x.

dx

d

d1

(csc x) =

dx

dx sin x

(sin x)(0) - (1)(cos x)

=

sin2 x

cos x

= - sin2 x

1 cos x =-

sin x sin x

= - csc x cot x

tan 5. Evaluate lim

3

+

h

- tan

3

by interpreting the limit as the derivative of a

h0

h

function at a particular point.

tan lim

3

+

h

- tan

3

d = (tan x)

= sec2 = 4

h0

h

dx

x=

3

3

For problems 6-14, differentiate

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

7. f (x) = 5 cos x + cot x -5 sin x - csc2 x

2

8. g(x) = 4 csc x + 2 sec x -4 csc (x) cot (x) + 2 sec (x) tan (x)

9. f (x) = sin x cos x cos2 x - sin2 x sin2 x

10. f (x) = cos x

2 sin x + sin x tan2 x 11. f (x) = x3 sin x

3x2 sin x + x3 cos x 12. f (x) = sec2 x + tan2 x

4 sec2 (x) tan (x) x + sec x

13. f (x) = 1 + cos x

1 + 2 tan x + cos x + sec (x) tan (x) + x sin x (1 + cos x)2 d2y

For problems 14-17, compute dx2 14. f (x) = tan x 2 sec2 x tan x 15. f (x) = sin x - sin x 16. f (x) = cos2 x 2 sin2 x - 2 cos2 x 17. f (x) = sin2 x + cos2 x 0

3

For problems 18-19, find all values of x in the interval [0, 2] where the graph of the given function has horizontal tangent lines.

18. f (x) = sin x cos x 3 5 7 , , , ; 44 4 4

19. g(x) = csc x

3 ,

22

20. 21.

cos x

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

at the

x

point where x = .

with plots

12 y = 2 x -

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

(1)

conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot,

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

inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d,

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

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

polyhedra_supported, polyhedraplot, rootlocus, semilogplot, setcolors, setoptions,

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

Consider the graphs of f (x) = 2 cos(x) and g(x) = 2 sin(x) shown below on the a dplot

sqrt

2

sin

x

,

x

=

0 ..

Pi 2

,

scaling

=

constrained

PLOT ...

(2)

interval 0, . b d plot sqrt 2 cos x , x = 0 .. Pi , scaling = constrained 22

PLOT ...

(3)

display a, b

1.4

1.2

1

0.8

0.6

0.4

0.2

0

3 5 3 7

16

8 16

4 16

8

16

2

x

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

4

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

4

f

= -1 and g

= 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.

4

22. A 15 foot ladder leans against a vertical wall at an angle of with the horizontal, as shown in the figure below. The top of the ladder is h feet above the ground. If the ladder is pushed towards the wall, find the rate at which h changes with respect to at the instant when = 30. Express your answer in feet/degree.

dh 15 3

3

=

ft/radian =

ft/degree

d 2

24

23. Use the Intermediate Value Theorem to show that there is at least one point in the interval (0, 1) where the graph of f (x) = sin x - 1 x3 will have a horizontal tangent 3 line.

f (x) = cos x - x2. Firstly, notice that f (x) is continuous for all x; therefore, it is continuous for all x in [0, 1]. Secondly, notice that f (0) = 1 > 0 and f (1) = cos (1) - 1 < 0. Thus, the Intermediate Value Theorem states there is at least one x0 in the interval (0, 1) with f (x0) = 0. In other words, there is at least one x0 in (0, 1) where f (x) will have a horizontal tangent line.

24. Multiple Choice: At how many points on the interval [-, ] is the tangent line to the graph of y = 2x + sin x parallel to the secant line which passes through the graph endpoints of the interval?

(a) 0 (b) 1 (c) 2 (d) 3 (e) None of these

C

5

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

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

Google Online Preview   Download