Homework Answers
Add Answer to:
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' :...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Note that f is twice differentiable on means that the second derivative f''(x) exists for each...
Note that f is twice differentiable on
means that the second derivative f''(x) exists for each x that is
an element of
Explain in full details!
Suppose f is a twice-differentiable function on (0,00) (i.e. f"(x) exists for every 1 € (0,0)), and A = sup \f(x)] = sup{]f(x)]: 0<x<00}, B = sup \f" (x)], C = sup 18" (2)]. O<I<0 0<<< O<I<00 If A+C< show that BP < 4AC. (0,00 We were unable to transcribe this image
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'...
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?)
differentiable function and there exists 0
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function,...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed
(2) Suppose that f and 9 are differentiable on an open interval I and that a...
(2) Suppose that f and 9 are differentiable on an open interval I and that a € R either belongs to I or is an endpoint of I. Suppose further that g and g' are never zero on I\{a} and that lim f(x) is of the form 0/0. (a) If there is an M ER such that f'(2)/'(x) < M for all x E I\{a}, prove that \$(r)/g(x) < M for all x € I\{a}. (b) Is this result true...
It is known that f :(0,2) + R is a differentiable function such that \f'(x) <...
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.
3. (a) (3 points) Write the definition of the derivative of a differentiable function f(x) at...
3. (a) (3 points) Write the definition of the derivative of a differentiable function f(x) at = a; (b) (7 points) using the definition of derivative as in (a), find the derivative of the function f(x) = Vx at a = 2. (c) EXTRA CREDIT (2 points): State the MEAN VALUE THEOREM (you can also draw a picture) and give its PHYSICAL interpretation in terms of INSTANTANEOUS and AV- ERAGE VELOCITIES.
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Note that f is twice differentiable on
means that the second derivative f''(x) exists for each x that is
an element of
Explain in full details!
Suppose f is a twice-differentiable function on (0,00) (i.e. f"(x) exists for every 1 € (0,0)), and A = sup \f(x)] = sup{]f(x)]: 0<x<00}, B = sup \f" (x)], C = sup 18" (2)]. O<I<0 0<<< O<I<00 If A+C< show that BP < 4AC. (0,00 We were unable to transcribe this image
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?)
differentiable function and there exists 0
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed
(2) Suppose that f and 9 are differentiable on an open interval I and that a € R either belongs to I or is an endpoint of I. Suppose further that g and g' are never zero on I\{a} and that lim f(x) is of the form 0/0. (a) If there is an M ER such that f'(2)/'(x) < M for all x E I\{a}, prove that \$(r)/g(x) < M for all x € I\{a}. (b) Is this result true...
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.
3. (a) (3 points) Write the definition of the derivative of a differentiable function f(x) at = a; (b) (7 points) using the definition of derivative as in (a), find the derivative of the function f(x) = Vx at a = 2. (c) EXTRA CREDIT (2 points): State the MEAN VALUE THEOREM (you can also draw a picture) and give its PHYSICAL interpretation in terms of INSTANTANEOUS and AV- ERAGE VELOCITIES.
Most questions answered within 3 hours.
-
A car drives over the crest of a hill of radius 120m with a
speed of...
asked 4 minutes ago -
Implementation of a MapReduce-style distributed word count
application
For this assignment, you can use any programming...
asked 11 minutes ago -
In females, the labia swells and the vagin-a lubricates during
which phase of the sexual response...
asked 42 minutes ago -
2. An item costs a retailer $200. If a 30 percent markup is
desired, what should...
asked 44 minutes ago -
Your client, Anita, is hurt in a car accident and comes to you
for some advice....
asked 56 minutes ago -
how many mL of 0.1050 M NaOH is needed to reach a pH of 3.74
when...
asked 51 minutes ago -
QUESTION
Memory retrieval that is easier when the person is in the same
psychological condition during...
asked 54 minutes ago -
The mean annual inflation rate in the UNited States over the
past 98 years in 3.37%...
asked 1 hour ago -
Design a class Holiday that represents a
holiday during the year. This class has three
private...
asked 1 hour ago -
Problem 1 (Logistic Regression and KNN). In this problem, we
predict Direction using the data Weekly.csv....
asked 1 hour ago -
What is the difference between VNTRs (Variable Number Tandem
Repeats) and STRs (Short Tandem Repeats) used...
asked 1 hour ago -
Fill in
Isotope: 15 O
1. Element name:
2. Atomic number:
3. Mass number:
4. Number...
asked 1 hour ago