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Give an example of a function of two variables whose limit at (0
,
0) does not exist, that is
lim
(
x,y
)
→
(0
,
0)
f
(
x, y
) does not exist. Explain also why the limit does not exist.
Find the first partial derivatives (fxandfy) of the following functionf(x, y) =∫xyh(s)ds.
Give an example of a function of two variables such that
f
(0
,
0) = 0 but
f
is NOT continuous
at (0
,
0). Explain why the function
f
is NOT continuous at (0
,
0).
If f(x)=sin-¹x. Show that (1-x²) f'x(x)-xf'(x)=0. HENCE ,prove that f^(n+2)(0)=n²f^n(0)
You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides as shown in the diagram: If 𝑙 = 200𝑚𝑚 and 𝑤 = 150𝑚𝑚 calculate: a) The value of x which will give the maximum volume b) The maximum volume of the box c) Comment of the value obtained in part b.
Evaluate the line integral ∫𝒖(𝑥, 𝑦, 𝑧) × ⅆ𝒓 𝐶 , where 𝒖(𝑥, 𝑦, 𝑧) = (𝑦 2 , 𝑥, 𝑧) and the curve 𝑪 is described by 𝒛 = 𝑦 = 𝑒 𝑥 with 𝑥 ∈ [0,1].
Suppose that a population yy grows according to the logistic model given by formula:
yy = LL
1 + AAee−kkkk .
a. At what rate is yy increasing at time tt = 0 ?
b. In words, describe how the rate of growth of yy varies with time.
c. At what time is the population growing most rapidly?
A cable is to be run from a power plant on one side of a river 900 meters wide to a factory
on the other side, 300 meters downstream. The cost of running the cable under the water is $5
per meter, while the cost over land is $4 per meter. What is the most economical route over
which to run the cable?
A cylindrical can is to be constructed to hold a fixed volume of liquid. The cost of the
material used for the top and bottom of the can is 3 cents per square inch, and the cost of the
material used for the curved side is 2 cents per square inch. Use calculus to derive a simple
relationship between the radius and height of the can that is the least costly to construct.
The partial fractions for
a²/(x(x²+a²)) are
#340153 on Dec 2023