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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
Consider the interesting curve below which is described by the equation ( ( )) 12 2 cosh sinh y 1 x − = + Determine an expression for dy y' dx = (your expression will contain both x and y functions). Use your calculations to answer questions 1 to 4 below. 1. The derivative with regards to x of ( ) 1 2 sinh y − is 2 2. Using the chain rule the derivative of ( ( )) 1 2 cosh sinh y − with regards to x is 3 3. The derivative of 2 1 x + is 2 4. The simplified version of dy y' dx = in terms of x and y is
standard form of f (x) = - x³ + x⁴ - 2x²
For questions 1 consider the Maclaurin Series expansion of
x = + + + +1 x x2 x3 .....
e
2! 3!
1. The coefficient of the term containing x3 in the Maclaurin series expansion of e3x
is 3
1−e3x
Suppose an area 𝑆 of a parabola 𝑦=𝑥2 illustrated in Figure 1.1 below is divided into four stripes 𝑆1,𝑆2,𝑆3, and 𝑆4 as shown in Figure 1.2 below; State the formulas to estimate the sum of the rectangles (𝑅1.3 and 𝑅1.4) under the parabolic region 𝑆 illustrated in Figures 1.3, 1.4 and the sum of areas of 𝑝 rectangles.
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.
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.
Give an example of a function of two variables whose first order partial derivatives exist at
(0, 0) (that is fx(0, 0) and fy(0, 0) both exist), but f is NOT differentiable at (0, 0).
Explain also why the function is NOT differentiable at (0, 0).
Find the slopes of the tangents to the curves obtained by slicing the surface
x^{2}e^{y}-yze^{x}=0 with planes x=1 and y=1 at the point (1, 1), using implicit
differentiation.
