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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
a manufacturer knows that if x goods are demanded on a particular week, the total cost and revenue function will be: c[x]=14+3x and R[x]=18x-2x^2 respectively. calculate the level of demand that will maximize profits. calculate the amount of profit that will be realized at this maximum point
Consider the interesting curve below
which is described by the equation
cosh (sinh-¹(v²))=√1+x² dy Determine an expression for y'= (your expression will contain both x and y dx functions). Use your calculations to answer questions 1 to 4 below.
1. The derivative with regards to x of sinh ¹ (y² is
2. Using the chain rule the derivative of cosh (sinh-¹(y²)) with regards to x is
3. The derivative of √√1+x² is
dy
4. The simplified version of y'= in terms of x and y is
dx
2
3
