Calculus Answers

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.

Describe geometrical meaning of indefinite integral. Write down

       some properties of indefinite integral.