Mathematics

Problem

Let

h(x)=(2x-1)\sqrt{x}

for

x>0

. The graph of

h

is shown in the

xy

–plane. (a)–(d) address properties of

h

at the point where

x=4

. (a) Derive an explicit expression for

h'(x)

. Show all algebraic steps. (b) Evaluate

h'(4)

and state its numerical value. (c) Determine the equation of the tangent line to the graph of

h

x=4

, writing your final answer in point–slope form. (d) Use the tangent line found in part (c) to approximate

h(4.2)

. Clearly indicate the value obtained and the substitution used.