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Is f continuous on the interval a b if f prime is continuous on the interval a b why or why not?
Wiki User
∙ 14y ago
Yes. You can make sense of this by referring to the definition of a derivative:
f'(x) = lim h goes to 0 of (f(x+h)-(fx))/h
As long as f(x+h) (as h goes to 0), and f(x) are defined so is f'(x). In fact, the only way f' is defined is if f(x) is defined.
Wiki User
∙ 14y ago
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What is the difference between mean value theorem of integration and Mean Value Theorem of differentiation?
The mean value theorem for differentiation guarantees the existing of a number c in an interval (a,b) where a function f is continuous such that the derivative at c (the instantiuous rate of change at c) equals the average rate of change over that interval. mean value theorem of integration guarantees the existing of a number c in an interval (a,b)where a function f is continuous such that the (value of the function at c) multiplied by the length of the interval (b-a) equals the value of a the definite integral from a to b. In other words, it guarantees the existing of a rectangle (whose base is the length of the interval b-a that has exactly the same area of the region under the graph of the function f (betweeen a and b).
Why in rolles theorem closed interval is used for contuinity and open interval is used for differentiability?
It requires that f(a)=f(b) where a and b are beginning and ending points. Also, it says there is a c between a and such that f'(c)=0. If f were not differentiable on the open interval, the statement f'(c)=0 would be invalid.
What is piecewise smooth function?
f is a piecewise smooth funtion on [a,b] if f and f ' are piecewise continuous on [a,b]
If f is continuous for all x and y and has two relative minima then f must have at least one relative maximum?
Yes.If you find 2 relative minima and the function is continuous, there must be exactly one point between these minima with the highest value in that interval. This point is a relative maxima.Think of temperature for example (it is continuous).
Define upper and lower sums?
Let P = { x0, x1, x2, ..., xn} be a partition of the closed interval [a, b] and f a bounded function defined on that interval. Then: * the upper sum of fwith respect to the partition P is defined as: U(f, P) = cj (xj - xj-1) where cj is the supremum of f(x)in the interval [xj-1, xj]. * the lower sum of f with respect to the partition P is defined as L(f, P) = dj (xj - xj-1) where dj is the infimum of f(x) in the interval [xj-1, xj].
What kind of continuous function can change sign but is never zero?
If the function is continuous in the interval [a,b] where f(a)*f(b) < 0 (f(x) changes sign ) , then there must be a point c in the interval a<c<b such that f(c) = 0 . In other words , continuous function f in the interval [a,b] receives all all values between f(a) and f(b)
Prove that if the definite integral is continuous on the interval ab then it is integrable over the interval ab sorry that I couldn't type the brackets over ab because it doesn't allow?
why doesn't wiki allow punctuation??? Now prove that if the definite integral of f(x) dx is continuous on the interval [a,b] then it is integrable over [a,b]. Another answer: I suspect that the question should be: Prove that if f(x) is continuous on the interval [a,b] then the definite integral of f(x) dx over the interval [a,b] exists. The proof can be found in reasonable calculus texts. On the way you need to know that a function f(x) that is continuous on a closed interval [a,b] is uniformlycontinuous on that interval. Then you take partitions P of the interval [a,b] and look at the upper sum U[P] and lower sum L[P] of f with respect to the partition. Because the function is uniformly continuous on [a,b], you can find partitions P such that U[P] and L[P] are arbitrarily close together, and that in turn tells you that the (Riemann) integral of f over [a,b] exists. This is a somewhat advanced topic.
What is the difference between mean value theorem of integration and Mean Value Theorem of differentiation?
The mean value theorem for differentiation guarantees the existing of a number c in an interval (a,b) where a function f is continuous such that the derivative at c (the instantiuous rate of change at c) equals the average rate of change over that interval. mean value theorem of integration guarantees the existing of a number c in an interval (a,b)where a function f is continuous such that the (value of the function at c) multiplied by the length of the interval (b-a) equals the value of a the definite integral from a to b. In other words, it guarantees the existing of a rectangle (whose base is the length of the interval b-a that has exactly the same area of the region under the graph of the function f (betweeen a and b).
What is an example of a function that is continuous on the interval a b for which the conclusion of the mean value theorem does not hold?
Let f(x)=abs(x) , absolute value of x defined on the interval [5,5] f(x)= |x| , -5 ≤ x ≤ 5 Then, f(x) is continuous on [-5,5], but not differentiable at x=0 (that is not differentiable on (-5,5)). Therefore, the Mean Value Theorem does not hold.
Why in rolles theorem closed interval is used for contuinity and open interval is used for differentiability?
It requires that f(a)=f(b) where a and b are beginning and ending points. Also, it says there is a c between a and such that f'(c)=0. If f were not differentiable on the open interval, the statement f'(c)=0 would be invalid.
What is piecewise smooth function?
f is a piecewise smooth funtion on [a,b] if f and f ' are piecewise continuous on [a,b]
When is the new iMacs coming out and when?
Suppose f is continuous on [a,b]. Then there is a point C in (a,b) such that f(c)=[f(a)+ f(b)]/2. Why is the statement false? Please help....
What is an example of a function that is continuous and bounded on the interval a b but does not attain its upper bound?
A function whose upper bound would have attained its upper limit at a bound. For example, f(x) = x - a whose domain is a < x < b The upper bound is upper bound is b - a but, because x < b, the bound is never actually attained.
What does it mean when f is differentiable at x equals 2?
what does it mean when f(x) is differentiable along an interval?it means that f is continuous along that domain. In other words, the curve f is smooth and does not break at any point along the interval.what does it mean when f(x) is differentiable at a point c?It means that f is continuous above the domain given by the interval that is an infinitesimally small distance from c. In other words the curve, f(x), is smooth and does not break along the differentially small interval given by c and at all of the values unimaginably close to c.what does it mean when the derivative of f(x) at c equals 2?It means that the instantaneous rate of change (slope) of f(x) at that point is equal to 2.what does it mean when the derivative of f(x) everywhere along an interval equals 2?It means that every single point along that interval has the same slope of 2. In other words, that interval yields a line with a slope of 2.
What do the notes BCE and F do on the Violin?
In music, the notes B to C and E to F are the only regular notes which have an interval of a minor second.
What are the notes to 'Defying Gravity' in 'Wicked'?
C-prime is the dominant note in the song "Defying Gravity" in the musical "Wicked."Specifically, two stanzas are Elphaba's contributions to "Defying Gravity" in "Wicked." The notes of the first stanza are the same as those of the second. The following lists the notes sung by Elphaba in each of her two stanzas on the soundtrack of the original Broadway cast:C-prime, d-prime, f-prime (5 in succession), g-prime;C-prime, d-prime, f-prime (3), g-prime (2);A-prime (2), g-prime, f-prime, e-prime, f-prime, d-double prime (2), c-double prime;C-prime, b flat-prime, a-prime, g-prime, f-prime;C-prime, b flat-prime, a-prime, g-prime, f-prime, e-prime, d-prime;C-prime, b, c-prime, b, c-prime, g-prime, c-prime, c-prime, g;C-prime, b, c-prime, a, g (2);C-prime, b, c-prime, g-prime, c-prime, b, c-prime, c-double prime, b-prime, g-prime, c-prime (2), d-prime (2), c-prime;E-prime (3), d-prime, c-prime, g, c-prime;E-prime (2), d-prime, c-prime, b, g, a-prime, g-prime.
What is the Mean Value Theorem in calculus?
There are several generalizations and extensions, but the basic Mean Value Theorem (MVT) by Joseph Louis LaGrange (1797) is this:Given a function f(x) that is continuous over a closed interval [a,b] and differentiable over the open interval (a,b), then there must exist a place in (a,b) such that the slope of the tangent to f(x) there will equal the slope of the secant over [a,b]. Algebraically stated, there must exist a number t within that interval, i.e., a
