Download Practice Problems for Mathematics Exam: Derivatives and Implicit Differentiation and more Study notes Calculus in PDF only on Docsity!
Practice Problems for TEST#
Find the Derivative:
e x
d
x e
dx
2
d
x x x
dx
2 x
d
x e dx
dx
d x
dx x
2
5 3
d x 3 x
dx x
sin( )
d
x e
dx
sin( ) cos( )
d
x x
dx
2
sec ( )
d
x
dx
cos( )
x
d
e x
dx
2
tan( )
x
d
x e x
dx
2
sin( )
d
x
dx
2 5 x
d
x e
dx
5
d
x
dx
3 5
d
x
dx
2
sin( )
x
d
e
dx
Piecewise Functions:
3
x sin( ) if 0
x
0 if 0
x
f x
x
, Find f '(0)
2
5x if 2
x +x if 2
x
f x
x
, Find f '(2)
2
3x if 1
x +x+1 if 1
x
f x
x
, Find f '(1)
2
2
x +3x+1 if 1
2x +x+2 if 1
x
f x
x
, Find f '(1)
Expect Students to find left and right derivatives for all of the above cases.
Implicit Differentiation:
2 3
y x 0, find
dy
dx
, when ( , x y ) ( 1, 1)
Applications of Derivatives:
For what values of x, are the tangent lines to the following functions horizontal?
3 2
f x x x x
sin( )
x
f x e
For what values of x are the following functions not differentiable?
f ( ) x sin( ) cot( ) x x
cos(2 )
x
f x x e
2
x
f x
x
Find the equation of the tangent line to the graph of the following functions at the given points.
f ( ) x cot( ) x , when
x
3
f ( ) x x , when x 0
(Students get confused when the tangent line crosses the graph.)
Question: Can there be two functions whose derivative is equal to 5 for all values of x? Explain why it cannot
happen or give an example.
- The volume V of a sphere of radius r is
3 4
3
V r
. Find the surface area of the sphere if S is the
instantaneous rate of change of the volume with respect to r.
- The mass of the part of a 5 meter long metal rod that lies between its left end and a point x meters to the
right is
2
m 10 x x kg. Find the linear density when x = 1m, x = 2m, and x = 3m. Where is the density
the highest?
- At time t = 0 seconds a diver jumps from a diving board that is 32 ft. above the water. The position
function is
2
s t ( ) 16 t 16 t 32 ft. at time t seconds. When does the diver hit water and what is his
velocity at impact?
- The graphs of two function are given below. Decide if each function has an inverse function. If the inverse
function exist, state the approximate domain and range of the inverse function. If no inverse exist, leave the
-1 -0.5 0.5 1 1.
x
1
2
fx
Does f(x)
have an
inverse?
_________
_
Range of
inverse___
_________
______
Domain of
inverse___
_________
_____
0.2 0.4 0.6 0.8 1 1.2 1.
x
1
2
fx
Does f(x)
have an
inverse?
_________
_
Range of
inverse___
_________
______
Domain of
inverse___
_________
_____
