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)

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.

1

For problems 6-16, differentiate

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

sin2 x 7. f (x) =

cos x 8. f (x) = x3 sin x

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

1 10. f (x) = tan x2

11. f (x) = sec 2x

12. f (x) = cos3 3x

13. f (x) = sin x

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

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

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

d

17. Use the following table to calculate g 2 sin x

dx

4

x=3

x f (x) f (x) g(x) g (x) 1 -2 -5 3 9 2 5 -3 4 -2 3 -1 6 7 -6 4 3 1 -2 5 54 7 1 8

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

cos x

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

at the

x

point where x = .

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

2

For problems 21-22, find all values of x in the interval [0, 2] where the graph of the given function has horizontal tangent lines.

21. f (x) = sin x cos x

22. g(x) = csc x

23. 24.

Use the Intermediate ValuewithTploths eorem to show that there is at least one point in the

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

(1)

1 3 conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot, interval (0, 1) where the graph of f (x) = sin x - x will have a horizontal tangent display, dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d,

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

line.

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) = a d plot sqrt 2 sin x , x = 0.. Pi , scaling = constrained 2 PLOT ...

interval 0, .

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

2

2 PLOT ...

2 sin(x) shown below on the

(2)

(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

25. 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.

3

26. 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

For problems 27-28, use the Second Derivative Test to determine the relative (local) extrema. Express each as an ordered pair (x, y).

27. f (x) = sin (3x) on [0, ] 28. f (x) = sec (3x) on [0, ] For problems 29-30, Find all absolute extrema of the given function on the given interval. 29. f (x) = cos x - sin x on the interval [-, ].

30. f (x) = tan x + sin x on the interval - , .

44

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