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Clac project
Clac project
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Answers
1
Evaluate the limit: limₓ→0 (sin x)/x
B) 1
D) Does not exist
A) 0
C) infinity
2
Newton developed his fluxion method around 1665–1666
True
False
3
Who used calculus to help explain the motion of planets?
C) Charles Babbage
D) Stephen Hawking
B) James Clerk Maxwell
A) Isaac Newton
4
What is the derivative of f(x) = 3x² + 2x − 5?
A) 6x + 2
C) 2x − 5
D) 6x − 2
B) 3x²
5
The trapezoidal sum is
A) ∑f((x_i+x_i-1)/2)∆x
B) ∑(f(x_i)+f(x_i-1))/2∆x
D) ∑f(x_i)∆x
C) ∑f(x_i-1)∆x
6
\lim_{x\to∞}\left(\frac{3x^2+2x+1}{2x^2-5x+7}\right)
B) 7/4
D) 4/3
A) 3/2
C) 2/3
7
Galileo Galilei developed the notation for differentials
False
True
8
Mr.Rands is the best math teacher
True
False
9
if f(x) = √((e^2x)+1) then dy/dx = (√2)/2
True
False
10
∫(e^x)/(1+e^x) dx
D) x- ln(1-e^x)+C
C) e^x + x+ C
B) ln(1+e^x)+C
A) ln((1/e^2)+1)+C
11
The phrase “fundamental theorem of calculus” was not used until the 19th century.
False
True
12
csc(7\pi/6)
D) -2/\sqrt{3}
C) -2
A) 2/\sqrt{3}
B) 2
13
What does the ∫ symbol introduced by Leibniz represent?
B) Sum of squares
A) Infinity
C) An integral
D) A derivative
14
Newton referred to derivatives as “fluxions.”
False
True
15
The Mean Value Theorem is named after
D) Gauss
B) Euler
A) Newton
C) Lagrange
16
What was Newton’s nationality?
D) Swiss
B) Dutch
C) English
A) German
17
cot(3\pi/4)
C) 1
D) \sqrt{2}
B) -1
A)-\sqrt{2}
18
What field benefited greatly from the use of calculus in the 18th century?
C) Astronomy
D) Political science
B) Biology
A) Architecture
19
if f(x) = (lnx)^2 then f"(√e) =
C) 2/√e
B) 2/e
A）1/e
D) 1/2√e
20
\int_0^2\left(x^2+3x\right)dx
A) 26/3
B) 3/26
D) 26/3
C) 36/2
21
Who is widely considered one of the co-founders of calculus?
C) Isaac Newton
A) Blaise Pascal
D) Euclid
B) Galileo Galilei
22
The term “limit” became central to calculus in which century?
D) 19th
A) 16th
B) 17th
C) 18th
23
arctan(-1)
C) -\pi/4
D) \pi
B) 5\pi/4
A) 3\pi/2
24
Calculus was first fully developed in Ancient Greece.
False
True
25
find dy/dx if f(x) = 2cos^2(5x)
A) -20sin(5x)
C) 20cos(5x)
D) -20sin(5x)cos(5x)
B) 4 cos(5x)
26
find ∫tanx dx
D) ln|secx|+C
A) ln|tanx|+C
C) -ln|sinx|+C
B) ln|cosx|+C
27
\int_0^1(2
x
+3)2
dx
A) 49/3
C) 31/3
D) 39/3
B) 4
28
f(x) = ln(x+4+e^-3) then f(0) = -2/5
False
True
29
The ∫ symbol for integration was introduced by Newton in the Principia.
True
False
30
The Fundamental Theorem of Calculus establishes the connection between differentiation and integration.
False
True
31
Which of the following is a core concept in integral calculus?
C) Tangent line
A) Slope
D) Gradient
B) Area under a curve
32
Derivatives describe the rate of change of a function.
True
False
33
y = x⁵·e^x, what is dy/dx
B) 5x⁴ + e^x
C) e^x(x⁵ + 5x⁴)
A) x⁵e^x
D) 5x⁴e^(x²)
34
f'(x) is ______
C) Newton's notation
B) Euler's notation
D) Leibniz's notation
A) Lagrange's notation
35
f(x)= e^{2x}\left(x^3+1\right) then g'(2) =
C) 21e4
A) 30e4
B) 24e4
D) 6e4
36
find y' if find y' if x^6+y^6=6
A) - y^5/x^5
C) x^5/y^5
D) y^5/x^5
B) -x^5/y^5
37
What is the slope of the tangent line to f(x) = x³ at x = 2?
B) 6
C) 8
D) 4
A) 12
38
If f(x) = sinh(x), what is f ''(x)?
A. cosh(x)
B. sinh(x) ✅
C. cosh(x) + sinh(x)
D. sinh(x) + cosh(x)
39
Who independently developed calculus around the same time as Isaac Newton?
B) Archimedes
D) Gottfried Wilhelm Leibniz
A) Pythagoras
C) René Descartes
40
Which mathematical symbol did Leibniz introduce?
B) √
A) ∑
D) π
C) ∫
