Chapter 2.5 Practice Problems - College of Arts and Sciences
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)
d 2. Use the definition of the derivative to show that (cos x) = - sin x
dx Hint: cos ( + ) = cos cos - sin sin
3. Use the quotient rule to show that d (cot x) = - csc2 x. dx
d 4. Use the quotient rule to show that (csc x) = - csc x cot x.
dx
tan 5. Evaluate lim
3
+
h
- tan
3
by interpreting the limit as the derivative of a
h0
h
function at a particular point.
For problems 6-14, differentiate
6. f (x) = 2 cos x + 4 sin x 7. f (x) = 5 cos x + cot x 8. g(x) = 4 csc x + 2 sec x 9. f (x) = sin x cos x
sin2 x 10. f (x) =
cos x 1
11. f (x) = x3 sin x 12. f (x) = sec2 x + tan2 x
x + sec x 13. f (x) =
1 + cos x d2y
For problems 14-17, compute dx2 14. f (x) = tan x 15. f (x) = sin x 16. f (x) = cos2 x 17. f (x) = sin2 x + cos2 x
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
19. g(x) = csc x
20. 21.
Compute an equation point where x = .
with plots
animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d,
(1)
of the line which is tangent to the graph of f (x) = 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,
cos x x
at the
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 d plot sqrt 2 sin x , x =0.. Pi , scaling =constrained 2
PLOT ...
(2)
interval 0, 2 . b d plot
sqrt
2
cos
x
,
x
=
0 ..
Pi 2
,
scaling
=
constrained
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
2
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.
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.
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
3
................
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