Question

Consider the function f left parenthesis x right parenthesis equals 3 x squared minus 5 x minus 2f(x)=3x2−5x−2 and complete parts​ (a) through​ (c). ​(a) find f left parenthesis a plus h right parenthesisf(a+h)​; ​(b) find startfraction f left parenthesis a plus h right parenthesis minus f left parenthesis a right parenthesis over h endfraction f(a+h)−f(a) h​; ​(c) find the instantaneous rate of change of f when aequals=44. ​(a) f left parenthesis a plus h right parenthesisf(a+h)equals= nothing ​(simplify your answer. do not​ factor.)

Answer 1

Given f(x)=3x²-5x-2

a) To find f(a+h) replace x with a+h in the given function. So,

f(a+h)=3(a+h)²-5(a+h)-2

=3(a²+2ah+h²)-5(a+h)-2 By using the formula (x+y)²=x²+2xy+y².

=3a²+6ah+3h²-5a-5h-2 By distributing property.

b) Similarly to find f(a) we need to replace x with a. So,

f(a)=3a²-5a-2

So, f(a+h)-f(h)= (3a²+6ah+3h²-5a-5h-2)-(3a²-5a-2)

=3a²+6ah+3h²-5a-5h-2-3a²+5a+2.

=6ah+3h^2-5h (All other terms has been cancel out)