1. Use the definition of the derivative to show that for any x>0 d/dx(square roo

Question

1. Use the definition of the derivative to show that for any x>0

d/dx(square root(3x)) = 3/(square root(3x))

2. Consider the function f(x)= 3x^2 + 5/ x+ 1

a. Determine the domain of f.

b. compute f'(x)

c. Find the critical points

3. Find the equation if the tangent to the curve f(x) = 3x^4 - 8x^3 - 6

4. Consider the function

F(x) = a sin (x) + b if x < 0

F(x) = x^2 + a  if 0 < x < and equal to 1

F(x) = b cos (2(pi)(x)) + a if x > 1

Find the values for and and b for which the function is always continuous

Note: could you please give detailed descriptions and step by step solutions. Thankyou very much to everyone who replies :) btw if doesn't matter if you answer all of them or not!

Explanation / Answer

4) For it to be continuous

asin x + b = x^2 + a at x =0

i.e. b = 0+ a

       b = a

and x^2 + a = b * cos(2(pi)x) + a at x=1

i.e 1 + a = b cos0 + a

      b =1

therefore a = b = 1

2) Domain is all numbers except x =0 for which 1/x is not defined

    f'(x) = 3 * 2 * x + 5*(-1/x^2) + 0

         = 6x -5/x^2

    Critical point is when f'(x) = 0

        6x^3 - 5 =0

     x = cuberoot(5/6).

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