Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 17
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41Worksheet # 12: Higher Derivatives and Trigonometric Functions
1. Calculate the indicated derivative:
(a)
(1)
( ) =
(b)
(5)
( ) = 2
+ 4
(c)
( )
( ) = 4
(d)
( )
( ) =
2. Calculate the first three derivatives of ( ) =
and use these to guess a general formula for
( ),
the -th derivative of .
3. Let ( ) = + 2 cos( ).
(a) Find all values of where the tangent line to
at the point (
( )) is horizontal.
(b) What are the largest and smallest values for the slope of a tangent line to the graph of ?
4. Differentiate each of the following functions:
(a)
( ) = cos( )
1
(b) ( ) =
cos( )
(c) ( ) =
sin( )
(d) ( ) = tan( ) + csc( )
(e)
( ) = sin( ) csc( )
(f)
( ) =
sin( )
(g) ( ) = sec( ) + cot( )
5. Calculate the first five derivatives of ( ) = sin( ). Then determine
and
6. Calculate the first 5 derivatives of ( ) = 1
. Can you guess a formula for the th derivative,
?
7. A particle’s distance from the origin (in meters) along the -axis is modeled by ( ) = 2 sin( )
cos( ),
where is measured in seconds.
(a) Determine the particle’s speed (speed is defined as the absolute value of velocity) at
seconds.
(b) Is the particle moving towards or away from the origin at
seconds? Explain.
3
(c) Now, find the velocity of the particle at time =
. Is the particle moving toward the origin or
2
away from the origin?
(d) Is the particle speeding up at
seconds?
8. Find an equation of the tangent line at the point specified:
(a)
=
+ cos( )
= 0
(b)
= csc( )
cot( )
=
(c)
=
sec( )
=
9. Comprehension check for derivatives of trigonometric functions:
(a) True or False: If
( ) =
sin( ), then ( ) = cos( ).
2
(b) True or False: If
is one of the non-right angles in a right triangle and sin( ) =
, then the
3
hypotenuse of the triangle must have length 3.
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