The Derivative in Graphing and Applications
Quiz yourself by thinking what should be in
each of the black spaces below before clicking
on it to display the answer.
Help!
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| Increasing (dfn) | f is increasing on (x1,x2),
if f(x2)>f(x1) given x2>x1
-slope will always be (+)
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| Increasing (thm) | Let f be differentiable on I,
if the f'(x)>0,
then f is increasing on I
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| Decreasing (dfn) | f is decreasing on (x1,x2),
if f(x1)>f(x2) given x2>x1
-slope will always be (-)
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| Decreasing (thm) | Let f be differentiable on I,
If f'(x)<0,
then f is decreasing on I
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| Constant (dfn) | f is constant on (x1,x2),
if(x1)=f(x2)
-slope will always be 0
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| Constant (thm) | Let f be differentiable on I,
If f'(x)=0,
then f is constant on I
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| Concave up (thm | Let f be twice differentiable on I,
If f''(x)>0,
then f is concave up on I
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| Concave down (thm) | Let f be twice differentiable on I,
If f''(x)<0,
then f is concave down on I
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| Relative Maximum | Let x0 Є I,
if f(x0)≥f(x)
-change from inc. to dec.
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| Relative Minimum | Let x0 Є I,
if f(x0)≤f(x)
-change from dec. to inc.
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| First Derivative Test | Find f'(x)
Solve
Number line
-change from inc. to dec.=Rel Max
-change from dec. to inc.=Rel Min
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| 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)
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| 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
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| Inflection Point (dfn) | if f changes the direction of its concavity at the point (x0,f(x0))
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| Absolute Maximum (dfn) | Let x0 Є I,
If f(x0)≥f(x) for ALL x Є I
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| Absolute Minimum (dfn) | Let x0 Є I,
If f(x0)≤f(x) for ALL x Є I
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| Relative Extrema (dfn) | -Occurs in the NEIGHBORHOOD of a point
-Not necessarily the highest or lowest on the curve
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| 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
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| Rectilinear Motion -(Velocity function) | Let s(t) be position function,
then s'(t)=v(t) is velocity function
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| Rectilinear Motion -(Acceleration) | Let s(t) be position function,
then, s''(t)=v'(t)=a(t) is acceleration function
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| Rectilinear Motion -(Speed) | Note: Velocity=Speed and Direction
Speed=|v(t)|
-no direction involved
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| Velocity -(inc with time) | v(t)>0
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| Velocity -(dec with time) | v(t)<0
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| Velocity -(Particle not moving) | v(t)=0
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| Acceleration -(Particle speeding up) | v(t) and a(t) have same signs
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| Acceleration -(Particle slowing down) | v(t) and a(t) have different signs
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| Newtons Method | Find R
Guess x1
Use tangent line to find x2
us tangent line to find x3
(ect.)
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| Newtons Method -(Equation) | xn+1= xn-(f(xn))/f'(xn)
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| 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)
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| 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
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