What is the answer for question 4. b) ? http://postimg.org/image/zft18m0fb/

justaguide | College Teacher | (Level 2) Distinguished Educator

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The function `g(x) = (-3x^3)/4 + 3x` . A tangent to the curve representing the function is a line that touches it at only one point.

A secant line joining two points representing x = a and x = b is a tangent to the curve when b tends to a. The slope of the tangent at x = a is given by the limit `lim_(h->0)(g(x+h)- g(x))/h`

For the function `g(x) = (-3x^3)/4 + 3x` , the slope of the tangent at x = 1 is given by:

`lim_(h->0)(g(1+h) - g(1))/h`

= `lim_(h->0)((-3*(1+h)^3)/4 + 3*(1+h) - ((-3*1^3)/4 + 3*1))/h`

= `lim_(h->0)((-3*(1+h^3 + 3h^2 + 3h))/4 + 3 + 3h +3/4 - 3)/h`

= `lim_(h->0)((-3-3h^3 - 9h^2 - 9h)/4 + 3 + 3h +3/4 - 3)/h`

= `lim_(h->0)((-3h^3 - 9h^2 - 9h)/4 + 3h)/h`

= `lim_(h->0)(-3h^2 - 9h - 9)/4 + 3`

= `-9/4 + 3`

= 0.75

The slope of the tangent to the curve represented by the function`g(x) = (-3x^3)/4 + 3x` at x = 1 is 0.75