for the function f, construct and simplify the difference quotient (f(x + h) - f(x))/h. 19) f(x)=3x² + 4x
for the function f, construct and simplify the difference quotient (f(x + h) - f(x))/h. 19) f(x)=3x² + 4x
Answer
Explanation:
Step1: Find f(x + h)
Substitute x + h into f(x): [ \begin{align*} f(x + h)&=3(x + h)^2+4(x + h)\ &=3(x^{2}+2xh+h^{2})+4x + 4h\ &=3x^{2}+6xh+3h^{2}+4x + 4h \end{align*} ]
Step2: Substitute f(x + h) and f(x) into the difference - quotient formula
[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(3x^{2}+6xh+3h^{2}+4x + 4h)-(3x^{2}+4x)}{h}\ &=\frac{3x^{2}+6xh+3h^{2}+4x + 4h - 3x^{2}-4x}{h}\ &=\frac{6xh+3h^{2}+4h}{h} \end{align*} ]
Step3: Simplify the expression
Factor out h from the numerator and cancel out h: [ \begin{align*} \frac{6xh+3h^{2}+4h}{h}&=\frac{h(6x + 3h+4)}{h}\ &=6x+3h + 4 \end{align*} ]
Answer:
$6x+3h + 4$