Justifications Guide Flashcards
f is continuous at x=A
limx->A- f(x) = limx->A+ f(x)=f(A)=L
f is differentiable at x=A
f is continuous at x=A and limx->A-f’(x) = limx->A+ f’(x)
f is increasing on the interval (A,B)
f’(x)>0 on the interval (A,B)
f is decreasing on the interval (A,B)
f’(x)<0 on the interval (A,B)
f has a critical point at x=A
f’(A) = 0 or undefined
f has a relative minimum at x=A
f’(x) changes from negative to positive at x=A
f has a relative maximum at x=A
f’(x) changes from positive to negative at x=A
f is concave up on the interval (A,B)
f’‘(x)>0 on the interval (A,B)
f is concave down on the interval (A,B)
f’‘(x)<0 on the interval (A,B)
f has an inflection point at x=A
f’‘(x) changes signs at x=A
f has an absolute minimum at x=A
f has a critical point at x=A and f(A) has the lowest value of all critical values and endpoints
f has an absolute maximum at x=A
f has a critical point at x=A and f(A) has the highest value of all critical values and endpoints
f(x)=k for some x on the interval [A,B]
f is continuous on [A,B] and k is is between the values of f(A) and f(B) by the intermediate value theorem (IVT)
f’(x) = k for some x on the interval [A,B]
f is differentiable on [A,B] and f(B)-f(A)/B-A = K by the mean value theorem (MVT)
a particle is at rest at t=k
velocity equals 0 at t=k v(k)=0
a particle changes direction at t=k
velocity changes signs at t=k
a particle is speeding up at t=k
the particles velocity and acceleration have the same sign
a particle is slowing down at t=k
the particles velocity and acceleration have opposite signs
a particle is moving away from the origin at t=k
the particles position and velocity have the same sign
a particle is moving towards the origin at t=k
the particles position and velocity have opposite signs
a tangent line approximation for f(A) is an underestimate
f is concave up at x=A
a tangent line approximation for f(A) is an overestimate
f is concave down at x=A
a right Riemann sum is an underestimate
f is decreasing on the interval
a right Riemann sum is an overestimate
f is increasing on the interval
a left Riemann sum is an underestimate
f is increasing on the interval
a left Riemann sum is an overestimate
f is decreasing on the interval
a trapezoidal approximation is an underestimate
f is concave down on the interval
a trapezoidal approximation is an overestimate
f is concave up on the interval
