Applications/motivation of derivatives:
Finding tangent lines (geometric)
Computing instant speed (physics)
[Plot the graph of position — time (vertical - horizontal)
at the point t= a, the slope of the tangent line to the graph is the instant speed at t=a. ]
Recall: tangent lines for circle.
a straight line L, and a circle C,
L and C can meet at 0, or 1, or 2 points.
L and C meeting at 1 point, is the case that L is a tangent line of C.
Question: Let us take the definition that a tangent line is a “limit” case of the secant lines.
Can we write down the equation of the tangent at a point 
)#card=math&code=%28a%2C%20f%28a%29%29)?
The equation of line has the form: 
or 
#card=math&code=y-y_0%20%3D%20k%20%28x-%20x_0%29), where 
#card=math&code=%28x_0%2C%20y_0%29) is a fixed given point.
What is the slope 
?
Answer: if the limit 
-f(a)%7D%7Bx-a%7D#card=math&code=%5Clim_%7Bx%20%5Cto%20a%7D%20%5Cfrac%7Bf%28x%29-f%28a%29%7D%7Bx-a%7D) exists, then it is the required slope.
The limit -f(a)%7D%7Bx-a%7D#card=math&code=%5Clim%7Bx%20%5Cto%20a%7D%20%5Cfrac%7Bf%28x%29-f%28a%29%7D%7Bx-a%7D) is a 
indefinite form. (e.g. ).
Tricks on 3.3 Exercises
)#card=math&code=3x%20%2818x%5E4%2B13%28x%2B1%29%29) using the derivative rule of products, or expand firstly.- similary as 111.
- first expand into the form
%20%3D%202%2B%205%2Fx#card=math&code=f%28x%29%20%3D%202%2B%205%2Fx). - easy
- %20%3D%204x%20-%202%2Fx%20%2B%201%2Fx%5E2#card=math&code=f%28x%29%20%3D%204x%20-%202%2Fx%20%2B%201%2Fx%5E2). [Better than direct computing using the derivative rule of quotients.]
- Write #card=math&code=f%28x%29) in the form that %3D%201%2B8%2F(x%5E2-4)#card=math&code=f%28x%29%3D%201%2B8%2F%28x%5E2-4%29).
Examples of Fermat’s theorem:
- %3Dx%5E2#card=math&code=f%28x%29%3Dx%5E2). At
, 
has a local minimum. We compute that 
%3D2x#card=math&code=f%27%28x%29%3D2x). And 
%3D0#card=math&code=f%27%280%29%3D0). 
%3D%5Csin(x)#card=math&code=y%28x%29%3D%5Csin%28x%29) . At 
#card=math&code=x%3D%20%5Cpi%2F2%2C%20y%28x%29) attains a local maximum. 
%3D%5Ccos%20(x)#card=math&code=y%27%28x%29%3D%5Ccos%20%28x%29). And 
%3D%5Ccos%20(%5Cpi%2F2)%3D0#card=math&code=y%27%28%5Cpi%2F2%29%3D%5Ccos%20%28%5Cpi%2F2%29%3D0).
Example of Statement iii of Theorem 4.9 (First derivative test):
%3Dx%5E3#card=math&code=f%28x%29%3Dx%5E3). 
. %3D3x%5E2#card=math&code=f%27%28x%29%3D3x%5E2). %3D0#card=math&code=f%27%280%29%3D0). But no matter 
, we always have %3D3x%5E2%20%3E0#card=math&code=f%27%28x%29%3D3x%5E2%20%3E0). Thus, is neither a local maximum nor a local minimum. This can also be explained by noting the monotonicity of 
: as 
increases, also increases.
