Section 1.2 The slope of a Tangent
The slope of a tangent to a curve at a point P is the limit of the slopes of the secants PQ as Q moves closer to P
mtangent=lim(Q -> P) (slope of PQ)
The slope of a tangent to a graph y=f(x) at the point (a, f(a)) is
mtangent= lim(∆x -> 0) \(\frac{∆y}{∆x}\)
= lim(h – >0)\(\frac{f(a+h)-f(a)}{h}\)
Section 1.3 The Rates of change
The average velocity (vector quantity, not to be confused with the speed ) is to be found the same way as that of a slope
\(Average\hspace{2mm} velocity =\frac{change\hspace{2mm} in \hspace{2mm} position}{change\hspace{2mm} in \hspace{2mm}time}\)
\(\bar{V}\)= \(\frac{∆s}{∆t}\)= \(\frac{s(a+h)- s(a)}{h}\)
The instantaneous velocity is the same as the slope of a tangent
v(a)=lim(h – >0)\(\frac{s(a+h)-s(a)}{h}\)
Section 1.4 The Limit of a function
The limit of a polynomial function y=f(x) at a point (a, f(a)) is
lim(x – >a)f(x) = f(a)
Section 1.5 Continuity at a point
The function is continuous at x=a if
(i) f(a) is defined
(ii)l lim(x – >a)f(x) exists and
(iii) lim(x – >a)=f(a)
Otherwise, it is discontinuous at f(a) if there is a break, hole, jump, or vertical asymptote.
A rational function h(x)= \(\frac{f(x)}{g(a)}\) is continuous at a if g(a)≠0