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FINAL Math 251 Practice
Note: This is a practice final and is intended only for study purposes. The actual exam will
contain different questions and may have a different layout.
- [] TRUE/FALSE: Circle T in each of the following cases if the statement is always
true. Otherwise, circle F. Let f and g be differentiable functions and h be a constant.
T F
x+h
2 x
1+h
x
T F (x + 1)
6 = x
6
5
4
3
2
T F
x
2
2 = x + h
T F lim x→r f (x) = f (r) for all r in the domain of f.
T F If lim x→r g(x) = 0, then lim x→r
f (x)
g(x)
does not exist.
T F
d
dx
1
x
Show your work for the following problems. The correct answer with
no supporting work will receive NO credit (this includes multiple choice
questions).
- Using the definition, find the derivative of f (x) =
4 x −
3
2
- Given that f (x) is a differentiable function and that a and k are constants, complete
the following Derivative Rules:
(a)
d
dx
(a
k ) = (b)
d
dx
(f (x)
k ) =
(c)
d
dx
(a
f (x) ) = (d)
d
dx
ln(f (x)) =
(e)
d
dx
(e
f (x) ) = (f )
d
dx
(log a
(f (x)) =
- Prove if f and g are differentiable, then
d
dx
(f − g) =
d
dx
f −
d
dx
g. Hint: use the definition
of a derivative.
- Find lim
x→ 0
x
4 sin(
x
). Recall that sin(
1
x
) oscillates between -1 and 1 as it gets closer to
zero. Explain your reasoning.
- Prove that the function f (x) = x
3 − 5 has a fixed point. (i.e. show that there exists a
point p such that f (p) = p)
- Given that x
2
2 = 25 find y
′
. Then use y
′ to linearly approximate the curve at
x = −3. Hint: you should find two linear approximations.
- Find
dy
dx
for each of the following:
y = x sin
1
x
y = x
x
x
x
2 y
2 = 4 − y arctan(5x) y =
xe
x
7
(x
6
10
- Given f (x) = e
− 3 x write down f
(n) (x) (the n
th derivative) for all n.
- Fill out the following and then graph f (x) = 2 cos x + sin 2x
- Domain:
- x-intercepts:
- y-intercepts:
- Symmetry of f (x):
- vertical asymptotes:
- horizontal asymptotes:
- extrema (both x and y coordinates)
- intervals f (x) is increasing:
- pts. of inflection (both x and y coordinates)
- intervals f (x) is concave up
- Graph f (x)
- A water tank has the shape of an inverted circular cone with base radius 2m and height
4m. If water is being pumped into the tank at a rate of 2m
3 /min, find the rate at which
the water level is rising when the water is 3m deep.
Recall the volume of a cone is
1
3
πr
2 h where r is the radius of the base and h is the
height of the cone.
