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Calculus- Chapter 3
The Derivative in Graphing and Applications
| 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 |
Created by:
KierstyN_O13
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