| Question | Answer |
| Increasing (dfn) | f is increasing on (x1,x2),
if f(x2)>f(x1) given x2>x1
-slope will always be (+) |
| Increasing (thm) | Let f be differentiable on I,
if the f'(x)>0,
then f is increasing on I |
| Decreasing (dfn) | f is decreasing on (x1,x2),
if f(x1)>f(x2) given x2>x1
-slope will always be (-) |
| Decreasing (thm) | Let f be differentiable on I,
If f'(x)<0,
then f is decreasing on I |
| Constant (dfn) | f is constant on (x1,x2),
if(x1)=f(x2)
-slope will always be 0 |
| Constant (thm) | Let f be differentiable on I,
If f'(x)=0,
then f is constant on I |
| Concave up (thm | Let f be twice differentiable on I,
If f''(x)>0,
then f is concave up on I |
| Concave down (thm) | Let f be twice differentiable on I,
If f''(x)<0,
then f is concave down on I |
| Relative Maximum | Let x0 Є I,
if f(x0)≥f(x)
-change from inc. to dec. |
| Relative Minimum | Let x0 Є I,
if f(x0)≤f(x)
-change from dec. to inc. |
| First Derivative Test | Find f'(x)
Solve
Number line
-change from inc. to dec.=Rel Max
-change from dec. to inc.=Rel Min |
| Second Derivative Test | find f''(x)
Solve
-f'(x0)=0 and f''(x0)<0,
then x0 is Rel Min (CD)
-f'(x0)=0 and f''(x0)>0,
then x0 has Rel Min (CU) |
| Graphing (steps) | 1. Domain
2. Asymptotes/End Behavior
-Limits
-if Dom is (-∞,+∞), there is no
VA
3. FDT- include Rel Max/Rel Min
4. "SDT" - CU/CD, IP
5. Intercepts and Symmetry
-set x=0 and y=0 |
| Inflection Point (dfn) | if f changes the direction of its concavity at the point (x0,f(x0)) |
| Absolute Maximum (dfn) | Let x0 Є I,
If f(x0)≥f(x) for ALL x Є I |
| Absolute Minimum (dfn) | Let x0 Є I,
If f(x0)≤f(x) for ALL x Є I |
| Relative Extrema (dfn) | -Occurs in the NEIGHBORHOOD of a point
-Not necessarily the highest or lowest on the curve |
| Absolute Extrema (thm) | Let f be continuous on [a,b],
then f has both an Abs Max and Abs Min on [a,b]
-Might occur on endpoints or critical points |
| Rectilinear Motion
-(Velocity function) | Let s(t) be position function,
then s'(t)=v(t) is velocity function |
| Rectilinear Motion
-(Acceleration) | Let s(t) be position function,
then, s''(t)=v'(t)=a(t) is acceleration function |
| Rectilinear Motion
-(Speed) | Note: Velocity=Speed and Direction
Speed=|v(t)|
-no direction involved |
| Velocity
-(inc with time) | v(t)>0 |
| Velocity
-(dec with time) | v(t)<0 |
| Velocity
-(Particle not moving) | v(t)=0 |
| Acceleration
-(Particle speeding up) | v(t) and a(t) have same signs |
| Acceleration
-(Particle slowing down) | v(t) and a(t) have different signs |
| Newtons Method | Find R
Guess x1
Use tangent line to find x2
us tangent line to find x3
(ect.) |
| Newtons Method
-(Equation) | xn+1= xn-(f(xn))/f'(xn) |
| Rolles Thm | Let f be cont. on [a,b] and diff on (a,b),
If f(a)=0 and f(b)=0,
then there exists at least one
c Є (a,b),
So that f'(c)=0
Conclusion: f'(c)=0 for c Є (a,b) |
| Mean-Value Thm | Let f be cont. on [a,b] and diff on (a,b)
Then there exists at least one
c Є (a,b)
Such that f'(c)=(f(b)-f(a))/b-a |
