Chapter 5 Flashcards
Differentiability-
Let g: A to R be a function defined on an interval A. Given c in A, the derivative of g at c is…
g’(c) = lim as x goes to c of
[g(x) - g(c)] / (x - c)
provided the limit exists.
g’(c) = lim as x goes to c of
[g(x) - g(c)] / (x - c)
provided the limit exists.
Differentiability-
Let g: A to R be a function defined on an interval A. Given c in A, the derivative of g at c is…
Algebraic Differentiability Theorem- Let f and g be functions defined on an interval A, and assume both are differentiable at some c in A, then…
i. ) (f + g)’(c) = f’(c) + g’(c)
ii. ) (kf)’(c) = k*f’(c) for k in R
iii. ) (f*g)’(c) = f’(c)*g(c) + f(c)*g’(c)
i. ) (f + g)’(c) = f’(c) + g’(c)
ii. ) (kf)’(c) = k*f’(c) for k in R
iii. ) (f*g)’(c) = f’(c)*g(c) + f(c)*g’(c)
Algebraic Differentiability Theorem- Let f and g be functions defined on an interval A, and assume both are differentiable at some c in A, then…
Chain Rule-
Let f: A to R and g: B to R satisfy f(A) c B so that the composition g(f(x)) is defined. If f is differentiable at c in A, and if g is differentiable at f(c) in B, then…
g(f(x)) is differentiable at c with
[g(f(c))]’ = g’(f(c))*f’(c)
g(f(x)) is differentiable at c with
[g(f(c))]’ = g’(f(c))*f’(c)
Chain Rule-
Let f: A to R and g: B to R satisfy f(A) c B so that the composition g(f(x)) is defined. If f is differentiable at c in A, and if g is differentiable at f(c) in B, then…
Interior Extremum Theorem-
Let f be differentiable on an open interval (a,b). If f attains a maximum (or minimum) value at some point c in (a,b)…
then f’(c) = 0.
then f’(c) = 0.
Interior Extremum Theorem-
Let f be differentiable on an open interval (a,b). If f attains a maximum (or minimum) value at some point c in (a,b)…
Darboux’s Theorem-
If f is differentiable on an interval [a,b] and if a’ satifies
f’(a) < a’ < f’(b), then…
there exists a point c in (a,b) where f’(c) = a’.
there exists a point c in (a,b) where f’(c) = a’.
Darboux’s Theorem-
If f is differentiable on an interval [a,b] and if a’ satifies
f’(a) < a’ < f’(b), then…
Rolle’s Theorem-
Let f: [a,b] to R be continuous on [a,b] and differentiable on (a,b). If f(a) = f(b), then…
there exists a point c in (a,b) where f’(c) = 0.
there exists a point c in (a,b) where f’(c) = 0.
Rolle’s Theorem-
Let f: [a,b] to R be continuous on [a,b] and differentiable on (a,b). If f(a) = f(b), then…
Mean Value Theorem-
If f: [a,b] to R is continuous on [a,b] an differentiable on (a,b), then…
there exists a point c in (a,b) where
f’(c) = [f(b) - f(a)] / (b - a)
there exists a point c in (a,b) where
f’(c) = [f(b) - f(a)] / (b - a)
Mean Value Theorem-
If f: [a,b] to R is continuous on [a,b] an differentiable on (a,b), then…
If g: A to R is differentiable on an interval A and satisfies g’(x) = 0 for all x in A, then…
g(x) = k for some k in R.
g(x) = k for some k in R.
If g: A to R is differentiable on an interval A and satisfies g’(x) = 0 for all x in A, then…
If f and g are differentiable on an interval A and
f’(x) = g’(x)
for all x in A, then…
f(x) = g(x) + k for some k in R.
f(x) = g(x) + k for some k in R.
If f and g are differentiable on an interval A and
f’(x) = g’(x)
for all x in A, then…
Generalized Mean Value Theorem-
If f and g are continuous on the closed inteval [a,b] and differentiable on the open interval (a,b), then…
there exists a point c in (a,b) where
[f(b) - f(a)]*g’(c) = [g(b) - g(a)]*f’(c)
or, if g’ is not 0,
f’(c) / g’(c) = [f(b) - f(a)] / [g(b) - g(a)]
there exists a point c in (a,b) where
[f(b) - f(a)]*g’(c) = [g(b) - g(a)]*f’(c)
or, if g’ is not 0,
f’(c) / g’(c) = [f(b) - f(a)] / [g(b) - g(a)]
Generalized Mean Value Theorem-
If f and g are continuous on the closed inteval [a,b] and differentiable on the open interval (a,b), then…
L’Hospital’s Rule for 0/0 Case-
Let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of a. If f(a) = g(b) = 0 and g’(x) is not 0 for all x not a, then
lim as x goes to a of
f’(x) / g’(x) = L
implies lim as x goes to a of
f(x) / g(x) = L.
lim as x goes to a of
f’(x) / g’(x) = L
implies lim as x goes to a of
f(x) / g(x) = L.
L’Hospital’s Rule for 0/0 Case-
Let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of a. If f(a) = g(b) = 0 and g’(x) is not 0 for all x not a, then
Given g: A to R and a limit point c of A, we say that limit x to c g(x) = infinity if…
for every M > 0, there exists a δ > 0 such that whenever
0 < |x - c| < δ
it follows that g(x) > M.
for every M > 0, there exists a δ > 0 such that whenever
0 < |x - c| < δ
it follows that g(x) > M.
Given g: A to R and a limit point c of A, we say that limit x to c g(x) = infinity if…
L’Hopital’s Rule for inf/inf Case-
Assume f and g are differentiable on (a,b) and that g’(x) is not 0 for all x in (a,b). If lim x goes to a of g(x) = infinity, then…
lim as x goes to a of
f’(x) / g’(x) = L
implies that lim x goes to a of
f(x) / g(x) = L.
lim as x goes to a of
f’(x) / g’(x) = L
implies that lim x goes to a of
f(x) / g(x) = L.
L’Hopital’s Rule for inf/inf Case-
Assume f and g are differentiable on (a,b) and that g’(x) is not 0 for all x in (a,b). If lim x goes to a of g(x) = infinity, then…
