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The temperature in the big hall is approximated by the function
T(x, y, z) = x
2 − 2xyz + z
2 + 5;
0 ≤ x ≤ 2, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 2.
If a person located at (1, 1, 1), in which direction he should walk
to cool off as rapidly as possibly.
Find the surface integral of the vector field 𝑭(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 𝑧) over the part of the paraboloid 𝑧 = 1 − 𝑥 2 − 𝑦 2 with 𝑧 ≥ 0 and having normal pointing upwards. Hint: take 𝑥 and 𝑦 as independent parameters
The temperature in the big hall is approximated by the function
T(x, y, z) = x
2 − 2xyz + z
2 + 5;
0 ≤ x ≤ 2, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 2.
If a person located at (1, 1, 1), in which direction he should walk
to cool off as rapidly as possibly.
Q.1: What are the applications of Calculus in engineering?
Q.1: Define differentiation and integration with example. What are the differences between them?
Q.3: Integrate the following functions with respect to x:
Sin3x, x^6 , xy, e^5x , 10 .
Q.4: Describe geometrical meaning of indefinite integral. Write down
some properties of indefinite integral.
a. Show that the curve with parametric equations
x=sint and y=sin(t+sint) for 0≤t≤2π
has two tangent lines at the origin and their equations. Illustrate by graphing the curve and its tangents.
b. Find the slope of the line tangent to the parametric curve
x=2cost and y=2-cos²t for 0≤t≤π
at points (-1, -1). Show the graph the parametric equations and the tangent line.
Use the principle of mathematical induction to show that
| sin nx| ≤ n| sin x|
for all n∈ N and for all x ∈ R
Evaluate the line integral
∫𝒖(𝑥, 𝑦, 𝑧) × ⅆ𝒓 ,
where 𝒖(𝑥, 𝑦, 𝑧) = (𝑦^2 , 𝑥, 𝑧) and the curve 𝑪 is described by 𝒛 = 𝑦 = 𝑒 𝑥 with 𝑥 ∈ [0,1].
Use differentials to approximate the volume of material needed to make a rubber ball if the radius of the hollow inner core is 2 in., and the thickness of the rubber is ⅛ in.
3.If y=80x-16x2, find the difference ∆y-dy if x=4 and ∆x=-0.2
In a right circular cone, the radius of the base is half as long as the altitude. If an error of 2% is made in measuring the radius, find the percentage of error made in the computed volume.
