Cumulative Review for the Final Exam
Math 1070
1. Use the graph to answer the following questions:
y
15
a) limx→1+ f (x) = b) limx→1− f (x) =
d) limx→5+ f (x) =
c) limx→1 f (x) =
e) limx→5− f (x) = f) limx→5 f (x) =
h) limx→8− f (x) =
g) limx→7 f (x) =
i) limx→8+ f (x) = j) a) limx→8 f (x) =
k) f (1) =
l) f (5) =
m) f (8) =
n) f (7) =
o) List all removable discontinuities
p) List all nonremovable discontinuities
q) List all points where f is not differentiable
10
5
x
0
1
2
3
4
5
6
7
8
2. Find the following limits:
3x3 − 3x + 9
3x5 − 3x + 9
b)
lim
a) lim
x→∞ 2x3 − 4x2 + 11
x→∞ 2x3 − 4x2 + 11
x2 − 3x
x2 − 9x
e) lim
d) lim 2
x→3 x − 9
x→3+ x2 − 9
√
3
(2 + h) − 8
x+6−3
h) lim
g) lim
x→3
h→0
h
x−3
3x3 − 3x + 9
x→∞ 2x4 − 4x2 + 11
x2 − 9x
f) lim
x→3− x2 − 9
1
x−1
x+1 − 2
i) lim
x→3 x − 3
c) lim
cos(x) − 1
. Check your answer using L’Hospital’s rule.
x→0
x2
3. Numerically evaluate the following limit lim
4. Find the following limits lim f (x), lim f (x), and lim f (x) and check if the function f is continuous,
x→2+
x→2−
x→2
½
4x − 1 if x ≤ 2
.
if f (x) =
x2 + 3 if x > 2
5. For the curve given below, sketch the derivative on the same axis.
y
2
y
1
5
2.5
0
0
0
0.5
1
1.5
2
2.5
-2.5
-1.25
0
1.25
2.5
x
x
-2.5
-1
-5
-2
(a)
(b)
6. Given f (x) = −x2 + 3x + 2. Using the definition of the derivative (limit process), find f 0 (x).
1
7. Find the equation of the tangent line to the graph of f (x) =
2x2
at the point where x = 1.
x+1
8. Given graph of the function f .
(i) Complete the table.
Point
A
B
C
D
E
Sign of f
Sign of f 0
Sign of f 00
(ii)At which points
(a) the value of f is largest
(b) the value of f is smallest
(c) the instantaneous rate of change of f is largest
(d) the instantaneous rate of change of f is smallest
(e) the instantaneous rate of change of f is positive
(f) the instantaneous rate of change of f is zero
(g) the instantaneous rate of change of f is negative
9. Compare the following two graphs and determine which one is graph of f and which is graph of f 0 .
y
y
6
10
4
5
2
0
0
0.25
0.5
0.75
1
x
-5
0
0
0.25
0.5
0.75
1
x
-10
-2
10. Let R(p) denote the revenue (in thousands of dollars) obtained for selling calculators at the price p
(in dollars). Explain the meaning of the following:
(a) R(75) = 625
(b) R(150) = 850
(c) R0 (75) = 10
(d) R0 (150) = −30
11. Differentiate the following
√ functions:
2
x + 3x + x
b) g(t) = t3 cos(2t)
a) f (x) =
x−1
√
d) h(y) = e−2y (y 3 + 4y + 1) e) g(x) = ln( x3 + 2)
√
y2 + 4
h) k(y) = 2
g) g(s) = s3 − s2 + 2
y −2
j) u(v) = v ln(v + 1)
2
2
c) f (x) = 43x
f) h(z) = z 2 sin(z√3 )
3t3 − 4 t
i) f (t) =
t2
12. Let f (1) = 3, g(1) = −2, f 0 (1) = 3, g 0 (1) = 2. Find
(a) h0 (1),where h(x) = f (x)g(x)
(b) k0 (1),where k(x) = f (x)/g(x)
(c) p0 (1),where p(x) = (f (x))2 + (g(x))3
13. Given the equation
Find
√
x y + y 3 − xy = 2x2 .
dy
using the implicit differentiation.
dx
14. Find (analytically) the global maximum and global minimum of the function on the given interval.
f (x) = x4 − 2x2 + 2 on the interval [−1, 3].
15. For the following function f (x) = 3x4 − 4x3 + 2 find (analytically):
(a)
i.
ii.
iii.
iv.
v.
vi.
vii.
Critical points
Intervals where f is increasing
Intervals where f is decreasing
Local minima and maxima
Intervals where f is concave up
Intervals where f is concave down
Inflection points
16. Sketch the possible graph of y = f (x) using given information. Assume that f is defined and
continuous everywhere.
(a)
x (−∞, a) a (a, b) b (b, c) c (c, ∞)
+
0
+
+
+
0
−
y0
00
y
−
0
+
0
−
−
−
(b)
x
y0
y 00
(−∞, a)
+
−
a
0
−
(a, b)
−
−
b
−
0
(b, c)
−
+
c
0
+
(c, ∞)
+
+
17. Consider a car which starts slowing down until completely stops. The car’s velocity is given in the
following table.
t [sec]
0
2
4
6
8 10
v(t) [ft/sec] 80 60 30 15 5 0
Find the left and right estimates for the distance traveled.
Z 3
ln(x)dx.Find the left and right sums using n = 20.
18. Given integral
1
19. A cylindrical container with no top is to be constructed to hold a fixed volume 144 in3 of liquid. The
cost of the material for the bottom is 80 cents/in2 and the cost for the material used for curved side
is 50 cents/in2 . Find the radius of the least expensive container.
3
20. A rectangular plot of land will be bounded on one side by a river and on the other three sides by a
single strand of electric fence. With 1,200 meters of wire at your disposal, what is the largest area
you can enclose?
Z 3
Z 3
Z 3
Z 3
Z 3
2
2
f (x)dx = 3;
g(x)dx = −1;
(f (x)) dx = 5;
(g(x)) dx = 3;
f (x)g(x)dx =
21. Let
1
1
1
1
Z 31
f (x)dx = −6 Find
−2;
Z 3
Z 1
Z 53
2
(cf (x) + dg(x))dx
b)
(f (x) + 2g(x)) dx c)
f (x)dx
a)
3
Z 3
Z 51
Z 13
(2f (x))2 − 3(
f (x)dx)2 e)
f (x)dx
d)
1
1
1
22. Evaluate
Z
1
1
a) (x3 − 3 √ + √
)dx
3
x
x
Z π
d)
(3 sin x − 2 cos x)dx
π/2
R2 5
3
1 ( x + x )dx
Z 2
√
y − 4 y + 6y − 2
e)
dy
y.
b)
c)
R3 1
2
1 ( x − 2x) dx
Z
1
f)
4ex dx
−1
23. Find the function f which satisfies given condition.
(a) f 0 (x) = 5ex − x2 , f (0) = 0.
24. Using
Z the method of
3
a) e−x x2 dx
Z
√
d) x x2 + 4dx
Rπ
g) 0 sin4 x cos xdx
substitution
find
Z
x
dx
b)
2
4
Z x −
x
e
dx
e)
x
3e − 2
R π sin x
dx
h) 0
2 + cos x
c)
f)
i)
Z
Z
R
1
x2
sin( x1 )dx
(ln x)3
dx
x
e2 sin(3x) cos(3x)dx
25. The graph of f is given. Sketch the graph of F such that F 0 (x) = f (x) and F (0) = 0.
2
y
y
6
5
1.5
4
1
3
2
0.5
1
0
0
2
4
6
0
8
0
x
-0.5
1
2
3
4
5
6
7
-1
-2
-1
-3
-4
-1.5
-5
-2
-6
Graph of f
(a)
(b)
(c)
(d)
(e)
(f)
Graph of F
Find x where F has local maximum.
Find x where F has local minimum.
Find F (1), F (2), F (4), F (6), F (8).
Find inflection points of F.
On the same graph sketch the graph of G such that G0 (x) = f (x) and G(0) = 1.
How G and F are related? Explain.
4
8
x