Calculus Answers

Trace the following curves
1)y=x^2/1-x^2
2)y=x+1/x

Expand sinx*coshx in ascending powers of x upto x^5

Find the value of 'a' for which the function f defined by
f(x)={(2-√(4-x))/x , x<0 ; a , x=0 ; (1-cosx)/2tan^(2)x, x>0
is continuous everywhere

Find the length of the arc of the hyperbolic spiral rθ = a lying between r = a and
r = 2a .

x − y + 2 = 0 is a tangent to the curve 3 2 (x + y) = (x − y + 2) at (−1, 1). true or false give reason also?

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = the quantity x minus seven divided by the quantity x plus three. and g(x) = quantity negative three x minus seven divided by quantity x minus one.

How can I transform a surge function (y=axe^-bx) horizontally and vertically? Does a surge function always have to go through the origin?

Let u = <-6, -2>, v = <-2, 3>. Find -3u + 2v

1. The polynomials ax^3+3x^2−13 and 2x^3 - 5x + a are divided by (x+2). If the remainder in each case is the same, find the value of a.

2. What must be subtracted from 4x^4 - 2x^3 - 6x^2 + x - 5 so that the result is exactly divisible by 2x^2 + x - 1? Use long division method.

3. What must be added to x^4 + 2x^3 - 2x^2 + x - 1 so that the result is exactly divisible by
x^2 + 2x - 3 ? Use long division method.

4. If a+b=1, then prove that a^3 + b^3 + 3ab = 1.

5. Factorise:
a) 2√2a^3 + 16√2b^3 + c^3 − 12abc
b) 2√2x^3 + 3√3y^3 + √5(5−5√6xy)

6. Factorise: a / b x^2 + (a / b + c / d)x + c / d; { b not equal to 0, d = 0 }

7. Factorise: (ax - by)^3 + (by - cz)^3 + (cz - ax)^3

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

8, -14, and 3 + 9i