Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Q1: Let f(t) be a real-valued function whose second derivative is positive for −∞ 
ltt
lt ∞. Which of the following statements is/are always true?      (2024)
(a) f(t) has at least one local minimum.
(b) f(t) cannot have two distinct local minima.
(c) f(t) has at least one local maximum.
(d) The minimum value of f(t) cannot be negative.
Ans: 
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)⇒ f(t) is a parabola open upwards. So, it has only one minima at its stationary point. 
∴ f(t) cannot have two distinct local minima.

Q2: Consider a vector Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)represent unit vector along the coordinate axes x, y, z respectively. The directional derivative of the function f(x, y, z) = 2 ln (xy) + ln(yz) + 3ln(xz) at the point (x, y, z) = (1, 1, 1) in the direction of Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(2024)
(a) 0
(b) 7/(5√2)
(c) 7
(d) 21
Ans:
(c)
Sol: We know that: directional derivative of 'f' at P(1, 1, 1) in the direction of   Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is given by
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
∴ Directional derivative Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q3: The closed curve shown in the figure is described by
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The magnitude of the line integral of the vector field Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) around the closed curve is ___(Round off to 2 decimal places).      2023Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(a) 9.42
(b) 6.36
(c) 2.45
(d) 7.54
Ans: 
(a)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q4: Consider the following equation in a 2-D realspace.
∣x1+ ∣x2= 1 for p > 0
Which of the following statement(s) is/are true.     (2023)
(a) When p = 2, the area enclosed by the curve is π.  
(b) When p tends to ∞, the area enclosed by the curve tends to 4.
(c) When p tends to 0 , the area enclosed by the curve is 1.
(d) When p = 1, the area enclosed by the curve is 2.
Ans: 
(a), (b), (d)
Sol: Check option (A),
put P = 2
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which is equation of circle whose radius is 1 .
∴ Area = πr= π
Check option (B),
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Curve :
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)∴ Area tends to 4 .
Check option (D),
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Curve:
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

∴ Area of curve (square) =(√2)= 2

Q5: One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation σ0.
The σis estimated by randomly drawing out 10,000 numbers of samples (xn). 
The estimates Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) are computed in the following two ways.
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which of the following statements is true?     (2023)
(a) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (c)
Sol: We know,
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Given, Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)We have,
Standard deviation, Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)and Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Here, we take samples from 106 numbers.
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q6: In the figure, the vectors u and v are related as: Au = v by a transformation matrix A. The correct choice of A is      (2023)
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(a) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(b) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (a)
Sol: Given :
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)For clockwise relation by θ, transformation matrix,
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)[∵ F or clockwise θ → N egative]

Q7: For a given vector w = [1 2 3]the vector normal to the plane defined by 𝑤𝑥=1wx = 1 is      (2023)
(a) [−2 −2 2]T
(b) [3 0 −1]T 
(c) [3 2 1]T 
(d) [1 2 3]T 
Ans:
(d)
Sol: Given, WT = 1
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)We have, vector normal to the plane = ∇F
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)∴ Normal vector = [1 2 3]T

Q8: Let Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) The value of Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) where V is the volume enclosed by the unit cube defined by Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) (2022)
(a) 3
(b) 8
(c) 10
(d) 5
Ans:
(c)
Sol: Divergence of Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Now,Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q9: Let Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) Then f(x) decreases in the interval     (2022)
(a) 𝑥(1,2)x ∈ (1, 2)
(b)  x ∈ (2, 3)
(c) x ∈ (0, 1)
(d) x ∈ (0.5, 1)
Ans: 
(a)
Sol: The function is decreasing, if f′(x) < 0
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)It is possible in between 1 & 2. Hence x ∈ (1, 2)

Q10: In the open interval (0, 1), the polynomial p(x) = x− 4x+ 2 has      (2021)
(a) two real roots
(b) one real root
(c) three real roots
(d) no real roots
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)It is clear that point of intersection of these graphs is solution (or) root of  p(x) = 0
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)According to intermediate value theorem
P(0) and P(1) are having opposite signs
∴ a root of p(x) = 0 in (0, 1)
and also from graph, there is only one point of intersection
Hence exactly one real root exists in (0, 1) .

Q11: Suppose the circles x+ y2 = 1 and (x − 1)+ (y − 1)= r2 intersect each other orthogonally at the point (u, v). Then u + v = _______.      (2021)
(a) 0
(b) 1
(c) 2
(d) 3
Ans: (b)
Sol:
If two curves cut orthogonally then product of slopes = −1
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q12: Let f(x) be a real-valued function such that f′(x0) = 0 for some x0 ∈ (0, 1), and f′′(x) > 0 for all  x ∈ (0, 1). Then f(x) has      (2021)
(a) no local minimum in (0,1)
(b) one local maximum in (0,1)
(c) exactly one local minimum in (0,1)
(d) two distinct local minima in (0,1)
Ans: 
(c)
Sol: x∈ (0, 1), where f(x) = 0 is stationary point
and f′′(x) > 0, ∀x ∈ (0,1)
So f′(x0) = 0  
and 𝑓′(0) > 0, where x∈ (0, 1)
Hence, f(x) has exactly one local minima in (0, 1)

Q13: Let 𝑎𝑥𝑎𝑛𝑑𝑎𝑦ax and ay be unit vectors along x and y directions, respectively. A vector function is given by F = axy − ayx  
The line integral of the above function Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)along the curve C, which follows the parabola y = x2 as shown below is _______ (rounded off to 2 decimal places).      (2020)
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(a) 2
(b) -2
(c) 3
(d) -3
Ans:
(d)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)= -3

Q14: If A = 2xi + 3yj + 4zk and u = x2 + y2 + z2, then div(uA) at (1, 1, 1) is____        (2019)
(a) 15
(b) 45
(c) 30
(d) 60
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q15: If f = 2x+ 3y2 + 4z, the value of line integral ∫cgrad f⋅dr evaluated over contour C formed by the segments (-3, -3, 2) → (2, -3, 2) → (2, 6, 2) → (2, 6, -1) is_______       (2019)
(a) 112
(b) 139
(c) 156
(d) 186
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q16: Let f(x) = 3x− 7x+ 5x + 6. The maximum value of f(x) over the interval [0, 2] is _______ (up to 1 decimal place).       (2018)
(a) 8.2
(b) 12.0
(c) 16.2
(d) 18.7
Ans: 
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)9x− 14x + 5 = 0
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Maximum [f(0), f′(0.55), f(2)]
Maximum [6, 7.13, 12] = 12

Q17: As shown in the figure, C is the arc from the point (3, 0) to the point (0, 3) on the circle x+ y2 = 9. The value of the integral  Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)  is _____ (up to 2 decimal places).      (2018)
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(a) 0
(b) 0.11
(c) 0.25
(d) 0.66
Ans:
(a)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)θ varies from 0 to π/2
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q18: Let f be a real-valued function of a real variable defined as f (x) = x-[x], where [x] denotes the largest integer less than or equal to x. The value of Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)is _______ (up to 2 decimal places).      (2018)
(a) 0.25
(b) 0.5
(c) 0.75
(d) 0.85
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q19: The value of the directional derivative of the function ϕ(x, y, z) = xy2+yz2+zx2 at the point (2,-1,1) in the direction of the vector p = i + 2j + 2k is       (2018)
(a) 1
(b) 0.95
(c) 0.93
(d) 0.9
Ans:
(a)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)THe directional derivative of f(x, y, z) at (2, -1, 1) in the direction of Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q20: Let f be a real-valued function of a real variable defined as f(x) = x2 for x > 0, and f(x) = −x2 for x < 0. Which one of the following statements is true?     (2018)
(a) f(x) is discontinuous at x = 0
(b) f(x) is continuous but not differentiable at x = 0
(c) f(x) is differentiable but its first derivative is not continuous at x = 0
(d) f(x) is differentiable but its first derivative is not differentiable at x = 0
Ans:
(d)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The first derivation of f (i.e) f'(x) is not derivable at x = 0.

Q21: LetPrevious Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Consider the composition of f and g, i.e., (f ∘ g)(x) = f(g(x)). The number of discontinuities in (f ∘ g)(x) present in the interval (−∞, 0) is:      (SET-2(2017))
(a) 0
(b) 1
(c) 2
(d) 4
Ans:
(a)
Sol: f(x) =1− x ; x< 0
g(x) = -x; x < 0
(Noth are continous for x < 0)
Therefore, fog(x) is continous for x < 0  
The composite function of two continous function is always continous . Therefore the number of discontinuities are zero.

Q22: Consider a function f (x, y, z) given by  f(x, y, z) = (x+ y− 2z2)(y+ z2) The partial derivative of this function with respect to x at the point, x = 2, y = 1 and z = 3 is ________      (SET-2(2017))
(a) 13
(b) 40
(c) 36
(d) 4
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q23: A function f(x) is defined as
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which one of the following statements is TRUE?
      (SET-1(2017))
(a) f(x) is NOT differentiable at x=1 for any values of a and b.
(b) f(x) is differentiable at x = 1 for the unique values of a and b
(c) f(x) is differentiable at x = 1 for all values of a and b such that a + b = e
(d) f(x) is differentiable at x = 1 for all values of a and b.
Ans: (b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)At x = 1,
L.H.S = R.H.S
e = a + b ...(i)
At x = 1, f'(x) exists
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)At x = 1,
L.H.S = R.H.S
e = 1 + 2a + b
2a + b = e - 1 ...(ii)
From equation (i) and (ii),
a = -1
b = e + 1
f(x) is differentiable at x = 1 for unique value of a and b.

Q24: Let Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) where R is the region shown in the figure and c = 6 × 10−4 The value of I equals________       (SET-1(2017))
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(a) 0.5
(b) 1
(c) 2
(d) 3
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q25: The value of the line integral Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is      (SET-2(2016))
(a) 0
(b) 2
(c) 4
(d) 6
Ans:
(b)
Sol: Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q26: The value of the integral Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) over the contour |z| = 1, taken in the anti-clockwise direction, would be      (SET-1(2016))
(a) Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b)Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) 24/13
(d) 12/13
Ans:
(b)
Sol: Singlarilies, Z = 1/2, 2 ± i
Only Z = 1/2 lies inside C
By residue theorem,
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Residue at 1/2
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q27: A function y(t), such that y(0) = 1 and y(1) = 3𝑒−1, is a solution of the differential equation Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Then y(2) is      (SET-1(2016))
(a) 5e−1 
(b) 5e−2
(c) 7e−2
(d) 7e−2
Ans: 
(b)
Sol: Auxiliary equation,
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q28: The maximum value attained by the function f(x) = x(x − 1)(x − 2) in the interval [1, 2] is _____.        (SET-1(2016))
(a) 0
(b) 1
(c) 2
(d) 4
Ans: (a)
Sol: 
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)stationary points are 1 + 1/√3
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Maximum value is 0.

Q29: The volume enclosed by the surface f(x, y) = ex over the triangle bounded by the lines x = y; x = 0; y = 1 in the xy plane is ______.       (SET-2(2015))
(a) 0
(b) 0.25
(c) 0.56
(d) 0.71
Ans:
(d)
Sol: 
Previous Year Questions- Calculus - 1 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

Q30: If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?     (SET-1 (2015))
(a) 𝑓(𝑎)𝑓(𝑏)=0f(a)⋅f(b) = 0
(b) f(a)⋅f(b) < 0
(c) 𝑓(𝑎)𝑓(𝑏)>0f(a)⋅f(b) > 0
(d) f(a)/f(b) ≤ 0
Ans:
(c)
Sol: Intermediate value theorem states that if a function is continious and 𝑓(𝑎)𝑓(𝑏)<0f(a)⋅f(b) < 0, then surely there is a root in (a, b). The contrapositive of this theorem is that if a function is continious and has no root in (a, b) then surely f(a)⋅f(b) ≥ 0. But since it is given that there is no root in the closed interval [a, b] it means 𝑓(𝑎)𝑓(𝑏)0f(a)⋅f(b) ≠ 0. 
So surely 𝑓(𝑎)𝑓(𝑏)>0f(a)⋅f(b) > 0 which is choise(C).

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FAQs on Previous Year Questions- Calculus - 1 - Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

1. What is the fundamental theorem of calculus?
Ans. The fundamental theorem of calculus states that if a function is continuous on a closed interval, then the definite integral of its derivative over that interval is equal to the difference of the function evaluated at the endpoints of the interval.
2. How can calculus be applied in electrical engineering?
Ans. Calculus is used in electrical engineering to analyze circuits, design control systems, and understand signals and systems. It helps in determining current and voltage relationships, solving differential equations in circuits, and optimizing system performance.
3. What are some common applications of calculus in electrical engineering?
Ans. Some common applications of calculus in electrical engineering include analyzing the behavior of electrical circuits, designing filters for signal processing, modeling electromagnetic fields, and optimizing power systems.
4. How can integration be used in electrical engineering?
Ans. Integration is used in electrical engineering to find the total charge or energy in a given system, calculate the power consumed by a device over time, and determine the total magnetic flux passing through a surface.
5. How does differentiation play a role in electrical engineering?
Ans. Differentiation is used in electrical engineering to find rates of change in voltage or current, determine the slope of a signal waveform, analyze the dynamic behavior of systems, and calculate the impedance of a circuit.
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