Download 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 0 (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
2. Use the definition of the derivative to show that
Hint: cos (α + β) = cos α cos β − sin α sin β
d
(cos x) = − sin x
dx
d
cos (x + h) − cos x
(cos x) = lim
h→0
dx
h
cos x cos h − sin x sin h − cos x
= lim
h→0
h
cos x cos h − cos x sin x sin h
= lim
−
h→0
h
h
cos h − 1
sin h
= lim cos x
− sin x
h→0
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
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
+ h − tan π3
by interpreting the limit as the derivative of a
5. Evaluate lim
h→0
h
function at a particular point.
tan π3 + h − tan π3
d
2 π
lim
=
(tan x)
= sec
=4
h→0
h
dx
3
x= π
tan
π
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
10. f (x) =
sin2 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) =
sin x
tan x + 3x
3x cos (x) − sec (x) tan (x) − 2 sin (x)
(tan x + 3x)2
13. f (x) = sec2 x + tan2 x
4 sec2 (x) tan (x)
14. f (x) =
x + sec x
1 + cos x
1 + 2 tan x + cos x + sec (x) tan (x) + x sin x
(1 + cos x)2
d2 y
For problems 15-18, compute
dx2
15. f (x) = tan x
2 sec2 x tan x
16. f (x) = sin x
− sin x
17. f (x) = cos2 x
2 sin2 x − 2 cos2 x
18. f (x) = sin2 x + cos2 x
0
3
For problems 19-20, find all values of x in the interval [0, 2π] where the graph of
the given function has horizontal tangent lines.
19. f (x) = sin x cos x
π 3π 5π 7π
, , ,
4 4 4 4
20. g(x) = csc x
π 3π
,
2 2
21. Compute an equation of the line which is tangent to f (x) =
1
2
y = 2x −
π
π
cos x
at x = π.
x
with plots
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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
Pi
a d plot sqrt 2 sin x , x = 0 ..
, scaling = constrained
2
PLOT ...
Pi
b d plot sqrt 2 cos x , x = 0 ..
, scaling = constrained
2
PLOT ...
display a, b
√
22. Considerh the igraphs of f (x) =
π
interval 0, .
2
2 cos(x) and g(x) =
(1)
√
2 sin(x) shown below on the
(2)
(3)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
π
16
π
8
3π
16
π
4
x
5π
16
3π
8
7π
16
π
2
π
Show that the graphs of f (x) and g(x) intersect at a right angle when x = . (Hint:
4
π
Show that the tangent lines to f and g at x = are perpendicular to each other.)
4
π π π
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.
4
23. A 15 foot ladder leans against a vertical wall at an angle of θ 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 θ
at the instant when θ = 30◦ . Express your answer in feet/degree.
√
√
dh
15 3
π 3
=
ft/radian =
ft/degree
dθ
2
24
5
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 0 (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
2. Use the definition of the derivative to show that
Hint: cos (α + β) = cos α cos β − sin α sin β
d
(cos x) = − sin x
dx
d
cos (x + h) − cos x
(cos x) = lim
h→0
dx
h
cos x cos h − sin x sin h − cos x
= lim
h→0
h
cos x cos h − cos x sin x sin h
= lim
−
h→0
h
h
cos h − 1
sin h
= lim cos x
− sin x
h→0
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
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
+ h − tan π3
by interpreting the limit as the derivative of a
5. Evaluate lim
h→0
h
function at a particular point.
tan π3 + h − tan π3
d
2 π
lim
=
(tan x)
= sec
=4
h→0
h
dx
3
x= π
tan
π
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
10. f (x) =
sin2 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) =
sin x
tan x + 3x
3x cos (x) − sec (x) tan (x) − 2 sin (x)
(tan x + 3x)2
13. f (x) = sec2 x + tan2 x
4 sec2 (x) tan (x)
14. f (x) =
x + sec x
1 + cos x
1 + 2 tan x + cos x + sec (x) tan (x) + x sin x
(1 + cos x)2
d2 y
For problems 15-18, compute
dx2
15. f (x) = tan x
2 sec2 x tan x
16. f (x) = sin x
− sin x
17. f (x) = cos2 x
2 sin2 x − 2 cos2 x
18. f (x) = sin2 x + cos2 x
0
3
For problems 19-20, find all values of x in the interval [0, 2π] where the graph of
the given function has horizontal tangent lines.
19. f (x) = sin x cos x
π 3π 5π 7π
, , ,
4 4 4 4
20. g(x) = csc x
π 3π
,
2 2
21. Compute an equation of the line which is tangent to f (x) =
1
2
y = 2x −
π
π
cos x
at x = π.
x
with plots
animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d,
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
Pi
a d plot sqrt 2 sin x , x = 0 ..
, scaling = constrained
2
PLOT ...
Pi
b d plot sqrt 2 cos x , x = 0 ..
, scaling = constrained
2
PLOT ...
display a, b
√
22. Considerh the igraphs of f (x) =
π
interval 0, .
2
2 cos(x) and g(x) =
(1)
√
2 sin(x) shown below on the
(2)
(3)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
π
16
π
8
3π
16
π
4
x
5π
16
3π
8
7π
16
π
2
π
Show that the graphs of f (x) and g(x) intersect at a right angle when x = . (Hint:
4
π
Show that the tangent lines to f and g at x = are perpendicular to each other.)
4
π π π
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.
4
23. A 15 foot ladder leans against a vertical wall at an angle of θ 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 θ
at the instant when θ = 30◦ . Express your answer in feet/degree.
√
√
dh
15 3
π 3
=
ft/radian =
ft/degree
dθ
2
24
5