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# Calculus- Chapter 3

### The Derivative in Graphing and Applications

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