æ
2
CMAS1
Name: ........................................................................................................
Date: ..........................................
1. Find local extremes and intervals where the function f (x) = 10x3 − 30x2 is
increasing, resp. decreasing.
Z √
4− x
+ 5 sin x dx.
x
2.
Evaluate the integral
3.
Evaluate the integral (by parts)
4.
Evaluate the integral (by substitution)
Z
(2x + 1)ex dx.
Z
cot 2x dx.
5. Find the 3rd order Taylor polynomial of the function f (x) = e2x at the point
a = 0.
3
RESULTS:
1.
f 0 (x) = 30x2 − 60x = 30x(x − 2),
f is increasing on (−∞, 0i, and h2, ∞), decreasing on h0, 2i,
lomax at x = 0, lomin at x = 2.
2.
√
√
Z x
4− x
4
+ 5 sin x dx =
−
+ 5 sin x dx =
x
x
Z x
√
1
1
4 · − x− 2 + 5 sin x dx = 4 ln |x| − 2 x − 5 cos x + c
x
Z 3.
f 0 = ex , f = ex ,
gZ = 2x + 1, g 0 = 2
Z
(2x + 1)ex dx = (2x + 1)ex − 2 ex dx = (2x + 1)ex − 2ex + c = (2x − 1)ex + c
4. Z
cot 2x dx =
1
2
Z
2 cos 2x
1
dx = ln | sin 2x| + c
sin 2x
2
5.
f 0 (x) = 2e2x , f 00 (x) = 2 · 2e2x , f 000 (x) = 2 · 2 · 2e2x ,
f (0) = 1, f 0 (0) = 2, f 00 (0) = 4, f 000 (0) = 8,
T3 (x) = 1 + 2x + 2x2 + 34 x3
4
CMAS2
Name: ........................................................................................................
Date: ..........................................
1. Find intervals of concavity and convexity and inflection points of the function
x−2
f (x) =
.
x
Z
2. Evaluate the integral (by parts) x ln x dx.
Z
3.
Evaluate the integral (by substitution)
Z
4.
Evaluate the definite integral
2
5.
4
cos4 x sin x dx.
1
dx.
x3
Find the first partial derivatives of the function f (x, y) = x5 − 3y 3 + 4x4 y 2 .
5
RESULTS:
1.
f 00 (x) = − x43 ,
f is convex on (−∞, 0), concave on (0, ∞),
no inflection points
2.
2
f 0 = x, f = x2 ,
g = ln x, g 0 = x1
Z
Z 2
x2
x 1
x2
x2
x ln x dx =
ln x −
· dx =
ln x −
+c
2
2 x
2
4
3.
t = cos x,
Zdt = − sin x dx
4
cos x sin x dx = −
Z
t4 dt = −
t5
cos5 x
+c=−
+c
5
5
4.
Z
4
2
4
2
2
2
3
1
dx = − 2 = − + =
x3
x 2
16 4
8
5.
∂f
∂x
∂f
∂y
= 5x4 + 16x3 y 2 ,
= −9y 2 + 8x4 y 3
6
CMAS3
Name: ........................................................................................................
Date: ..........................................
1. Find local extremes and intervals where the function f (x) =
resp. decreasing.
Z x − 3x2
√
+ 5 sin x
x
2.
Evaluate the integral
3.
Evaluate the integral (by parts)
x−1
is increasing,
x2
dx.
Z
(x − 1) sin x dx.
Z
4.
Evaluate the integral (by substitution)
1
dx.
5 − 2x
5. Find the 3rd order Taylor polynomial of the function f (x) = cos 3x at the
point a = 0.
7
RESULTS:
1.
f 0 (x) = 2−x
x3 ,
f is decreasing on (−∞, 0), and h2, ∞), increasing on (0, 2i,
lomax at x = 2.
2.
x − 3x2
√
+ 5 sin x
x
x2
x
√ − 3 √ + 5 sin x dx =
dx =
x
x
Z √
√
3
1
2
6
x3 −
x5 − 5 cos x + c
x 2 − 3x 2 + 5 sin x dx =
3
5
Z Z 3.
f 0 = sin x, f = − cos x,
gZ = x − 1, g 0 = 1
Z
(x − 1) sin x dx = −(x − 1) cos x + cos x dx = (1 − x) cos x + sin x + c
4. Z
1
1
dx = −
5 − 2x
2
Z
−2
1
dx = − ln |5 − 2x| + c
5 − 2x
2
5.
f 0 (x) = −3 sin 3x, f 00 (x) = −9 cos 3x, f 000 (x) = 27 sin 3x,
f (0) = 1, f 0 (0) = 0, f 00 (0) = −9, f 000 (0) = 0,
T3 (x) = 1 − 92 x2
8
CMAS4
Name: ........................................................................................................
Date: ..........................................
1. Find intervals of concavity and convexity and inflection points of the function
f (x) = 2x4 − 12x2 .
Z
2.
Evaluate the integral (by parts)
ln x dx.
3.
Evaluate the integral (by substitution)
Z
Z
4.
Evaluate the definite integral
3
6x2 ex dx.
2
(6x2 − 1) dx.
1
5.
Find the first partial derivatives of the function f (x, y) = 2x5 y 2 + 3x2 − 2y 3 .
9
RESULTS:
1.
f 00 (x) = 24x2 − 24 = 24(x + 1)(x − 1),
f is convex on (−∞, −1i and h1, ∞), concave on h−1, 1i,
inflection at x = −1 and x = 1
2.
f 0 = 1, f = x,
g = ln x, g 0 = x1
Z
Z
ln x dx = x ln x −
dx = x ln x − x + c
3.
t = x3 ,
Zdt = 3x2 dx
Z
3
2 x3
6x e dx = 2 et dt = 2et + c = 2ex + c
4.
Z
2
1
2
(6x2 − 1) dx = 2x3 − x 1 = (16 − 2) − (2 − 1) = 13
5.
∂f
∂x
∂f
∂y
= 10x4 y 2 + 6x,
= 4x5 y − 6y 2