Facts
If a function is differentiable it is . . .
For IVT to apply, f'(x) must be _________ on a _____ interval.
continous / closed
Crititcal points occur when what is true about f(x)?
f(x)=0 or f(x) is undefined
A point of inflection occurs on f(x) when . . .
f"(x) changes sign
f'(x) changes from increasing to decreasing
f'(x) changes from decreasing to increasing
If x(t) is the position function of an object, how do you find the objects velocity function v(t)?
v(t) = s'(t)
If x(t) is the position function of an object, how do you find the object's acceleration function a(t)?
a(t)=s"(t)
f(x) has a horizontal tangent line where . . .
f'(x)=0
For MVT to apply, f'(x) must be _______ on a/an ______ interval and ________ on a/an ________ interval.
continuous / closed
differentiable / open
If f(x) has a change in concavity, it will occur at a/an _____________.
point of inflection or inflection point
If a f(x) has an extrema, it will occur at a _______ _____.
critical point
If v(5)>0, then an object is moving which direction at t=5?
If v(5)<0, then an object is moving which direction at t=5?
f(x) has a vertical tangent line where . . .
f'(x) is undefined
If f(x) is continuous on [a,b] and f(a)<0 and f(b)>0, what do we know about a c value such that a<c<b?
f(a)<f(c)<f(b)
If f'(x) is positive, f(x) is ________
increasing
If f'(x) is negative, f(x) is __________.
decreasing
If v(5)<0 and a(5)<0, then an object is ______________ at t=5.
speeding up
If v(5)>0 and a(5)<0, then an object is ______________ at t=5.
slowing down
f(x) is continuous if . . .
lim_(x->c)f(x)=f(c)
If f(x) is continous on [a,b], we know f(x) have both an _______ ________ and an ________ _________.
bsolute minimum / absolute maximum
If f"(x) is negative, then f'(x) is ________ and f(x) is ______.
decreasing / concave down
If f"(x) is positive, then f'(x) is _________ and f(x) is ___________.
increasing / concave up
miles per hour per hour
m/h^2
If v(t) represents the velocity of an object, what does this integral represent?
int abs(v(t))dt
Total distance traveled
What must be true about the above limit for L'Hopital's Rule to apply? (I'm looking for the free response answer.)
lim_(x->c)f(x)/g(x)
lim_(x->c)f(x)=0 and lim_(x->c)g(x)=0
If f(x) is continuous on [a,b] and differentiable on (a,b), then, for some a<c<b, what must be true?
f'(c)=(f(b)-f(a))/(b-a)
f(x) has a relative min when . . .
f(x) changes from decreasing to increasing
f'(x) changes from negative to positive
f(x) has a relative max when . . .
f(x) changes from increasing to decreasing
f'(x) changes from positive to negative
If f'(t) represent the rate of change of an object in feet per second, the solution to this integral will have what units?
intf'(x)dx
feet
If a(t), the acceleration function, of an object is given, and an initial condition for v(t), its velocity, is given, explain how to find the v(t) function.
Integrate a(t) to find v(t), then use the initial condition for v(t) to solve for c. Finally, plug in the value for c and write the v(t) function.