To estimate the sloep of the tangent line we first need to find an expression for the estimation, we know that this can be done by:
[tex]m=\frac{f(x+h)-f(x)}{h}[/tex]Then we have:
[tex]\begin{gathered} m=\frac{\lbrack6.8(x+h)^2-3.4(x+h)\rbrack-\lbrack6.8x^2-3.4x\rbrack}{h} \\ =\frac{\lbrack6.8(x^2+2xh+h^2)-3.4x-3.4h\rbrack-\lbrack6.8x^2-3.4x\rbrack}{h} \\ =\frac{(6.8x^2+13.6xh+6.8h^2-3.4x-3.4h)-(6.8x^2-3.4x)}{h} \\ =\frac{13.6xh+6.8h^2-3.4h}{h} \\ =\frac{h(13.6x+6.8h-3.4)}{h} \\ =13.6x+6.8h-3.4 \end{gathered}[/tex]hence the estimation of the slope tangent line is:
[tex]m=13.6x+6.8h-3.4[/tex]To determine the slope at the given point for the value of h we just plug the values in the expression we found. for example for the first estimate we have:
[tex]m=13.6(7)+6.8(1)-3.4=98.6[/tex]doing this with all the other values of h we have:
h=1: m=98.6
h=0.5: m=95.2
h=0.1: m=92.48
h=0.01: m=91.868
h=0.001: m=91.8068
