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Quiz

Lesson 1 Quiz: Integration Techniques (Substitution and By Parts) and Definite Integrals

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  • Questions count: 20
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1.
STEM Calculus Problem 26: Find the derivative dy/dx for the function y = sin3(2x).
3 sin2(2x)
6 sin2(2x) cos(2x)
3 sin2(2x) cos(2x)
6 sin2(2x)
2.
STEM Integration Problem 43: Evaluate the definite integral ∫[0 to 2] (3x2 - 4x + 1) dx.
4
6
0
2
3.
STEM Integration Problem 24: Evaluate the definite integral ∫[0 to 2] (3x2 - 4x + 1) dx.
2
0
4
6
4.
STEM Algebra Problem 26: In the expansion of (x + 1/x)10, find the value of the term independent of x (the constant term).
252 (the 6th term T_6 = 10C5)
1
120
210
5.
STEM Limits Problem 6: Evaluate the limit: lim [x -> 0] (sin(5x) / tan(2x)).
5/2 (or 2.5)
1
0
Undefined
6.
STEM Integration Problem 11: Evaluate the definite integral ∫[0 to 2] (3x2 - 4x + 1) dx.
2
6
0
4
7.
STEM Statics Problem 14: A body of weight 60 N rests on a rough horizontal plane. If the coefficient of static friction μ_s = √3 / 3, what horizontal force is required to place the body on the verge of motion?
20√3 N (approximately 34.64 N)
30 N
60 N
10√3 N
8.
STEM Calculus Problem 25: Find the derivative dy/dx for the function y = sin3(2x).
6 sin2(2x)
3 sin2(2x) cos(2x)
3 sin2(2x)
6 sin2(2x) cos(2x)
9.
STEM Limits Problem 10: Evaluate the limit: lim [x -> 0] (sin(5x) / tan(2x)).
0
Undefined
1
5/2 (or 2.5)
10.
STEM Statics Problem 37: A body of weight 60 N rests on a rough horizontal plane. If the coefficient of static friction μ_s = √3 / 3, what horizontal force is required to place the body on the verge of motion?
20√3 N (approximately 34.64 N)
10√3 N
30 N
60 N
11.
STEM Limits Problem 15: Evaluate the limit: lim [x -> 0] (sin(5x) / tan(2x)).
1
0
Undefined
5/2 (or 2.5)
12.
STEM Algebra Problem 20: In the expansion of (x + 1/x)10, find the value of the term independent of x (the constant term).
1
252 (the 6th term T_6 = 10C5)
210
120
13.
What is the rigorous mathematical principle for: Fundamental Theorem of Calculus and Definite Integrals?
Definite integration always yields a family of functions with an arbitrary constant C
The integral of 1/x is strictly x2 / 2
Integration by parts states that ∫ u dv = u v + ∫ v du
If f is continuous on [a, b] and F is an antiderivative of f, then ∫[a, b] f(x) dx = F(b) - F(a), representing net accumulated area under the curve
14.
STEM Limits Problem 9: Evaluate the limit: lim [x -> 0] (sin(5x) / tan(2x)).
0
Undefined
1
5/2 (or 2.5)
15.
STEM Solid Geometry Problem 4: What is the equation of a sphere in 3D Cartesian coordinates with center C(2, -1, 3) and radius r = 5?
(x - 2)2 + (y - 1)2 + (z - 3)2 = 5
(x + 2)2 + (y - 1)2 + (z + 3)2 = 25
(x - 2)2 + (y + 1)2 + (z - 3)2 = 25
x2 + y2 + z2 - 4x + 2y - 6z = 25
16.
STEM Limits Problem 21: Evaluate the limit: lim [x -> 0] (sin(5x) / tan(2x)).
Undefined
5/2 (or 2.5)
1
0
17.
STEM Integration Problem 4: Evaluate the definite integral ∫[0 to 2] (3x2 - 4x + 1) dx.
4
6
2
0
18.
STEM Algebra Problem 29: In the expansion of (x + 1/x)10, find the value of the term independent of x (the constant term).
210
1
120
252 (the 6th term T_6 = 10C5)
19.
The common ratio r of the geometric sequence (64, 32, 16, 8, ...) is:
r = 2
r = 4
r = -1/2
r = 1/2
20.
STEM Calculus Problem 11: Find the derivative dy/dx for the function y = sin3(2x).
6 sin2(2x) cos(2x)
3 sin2(2x)
6 sin2(2x)
3 sin2(2x) cos(2x)
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