Thread: linear appx/graph reading

ok, I am completely at a loss - I don't even know where to begin.
for (a): I should find the slope of T - but I have no idea how - could I make a triangle and solve for the hyp? so would it be like $\displaystyle f'(a) = \sqrt{PQ^2 + QT^2}$

and then I am at a complete loss, partially because the given diagram is so confusing ... any help would be greatly welcomed

Originally Posted by cassiopeia1289

ok, I am completely at a loss - I don't even know where to begin.
for (a): I should find the slope of T - but I have no idea how - could I make a triangle and solve for the hyp? so would it be like $\displaystyle f'(a) = \sqrt{PQ^2 + QT^2}$

and then I am at a complete loss, partially because the given diagram is so confusing ... any help would be greatly welcomed

For a): The slope of the tangent line is equal to $\displaystyle \frac{QT}{PQ}$.

For b): The value of the function is the length of the segment $\displaystyle RS$.

any hints as to how to answer c and d?
those are where I am absolutely at a loss -
c: I mean, would you find like $\displaystyle {f(h) - f(a)}/(h-a)$??
and what are they even asking for d - and actual number?

Originally Posted by cassiopeia1289

any hints as to how to answer c and d?
those are where I am absolutely at a loss -
c: I mean, would you find like $\displaystyle {f(h) - f(a)}/(h-a)$??
and what are they even asking for d - and actual number?

For part c, you use the tangent line at $\displaystyle (a, f(a))$ to approximate the value of the function at $\displaystyle a + h$. In the drawing you have, this approximation is given by the length of $\displaystyle ST$. But the formula for this approximation is $\displaystyle y = f(a) + f'(a)(x - a)$, which simplifies to $\displaystyle f(a) + f'(a)\cdot{h}$ in this case.

For part d, this is simply $\displaystyle \frac{f(a + h) - f(a)}{h}$. It doesn't help you a whole lot except that as h approaches zero, this is the derivation of the derivative of $\displaystyle f(x)$ at a.