Let f: Rn R. Suppose that for each unit vector u Rn, the directional derivative

Question:

Let f: Rn → R. Suppose that for each unit vector u ∈ Rn, the directional derivative Duf(a + tu) exists for t ∈ [0, 1 ] (see Definition 11.19). Prove that
f(a + u) - f(a) = Duf(a + tu)
for some t ∈ (0, 1).

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

See More Books

Question Posted: Mar 29, 2016 04:21 AM

See More Questions

Let f : R x R R be differentiable. For each x R defined gx: R R By gx (y) = f(x,y). Suppose that for each x there is a unique y with gx1(y) =0; let c(x) be this y.
Let E be closed and bounded in R, and suppose that for each x E there is a function fx, nonnegative, nonconstant, increasing, and C on R, such that fx(x) > 0 and fx(y) = 0 for y E. Prove that there...
Let V be open in Rn, a V, f : V R, and suppose that f is differentiable at a. a) Prove that the directional derivative Duf(a) exists (see Exercise 11.2.10) for each u Rn such that ||u|| = 1 and...
How does Apple Inc apply Marketing and Advertising To market their product? Direct advertising Indirect Advertising Social Media (Paid Media, Owned Media, Earned Media)
J. Los Clothiers has forecast credit sales for the fourth quarter of the year as: September (actual)......................$70,000 Fourth Quarter October ....................................$60,000...
The head of research and development for the U.S. Army must select for procurement one of the antitank weapons from the accompanying listing. The listed results indicate the distance away from the...
Define the common barriers to effective communication.(pp. 132134)
On February 14, 2016, Isabelle Moretti, Aida Kam, and Channade Fenandoe start a partnership to operate a marketing consulting practice. They sign a partnership agreement to split profits in a 2:3:4...
David Morning, age 67, is retired and receives Social Security of $2,000 per month, and a pension of 1,100 per month. The pension distribution is from a qualified Code Sec. 401 (k) plan that Paul...
Picasso Company is a wholesale distributor of packaging equipment and supplies. The company?s sales have averaged about $900,000 annually for the 3-year period 2018?2020. The firm?s total assets at...
Suppose that V is open in Rn, that f: V R is C2 on V, and that fxj(a) = 0 for some a H and all j = I, ...,n. Prove that if H is a compact convex subset of V, then there is a constant M such that for...
For each of the following functions, prove that f-1 exists and is differentiable in some nonempty, open set containing (a, b), and compute D(f-1){a, b) a) f(u, v) = (3u - v, 2u + 5v) at any (a, b) ...
A. Determine the market price of an 6.75%, $50,000, 20-year coupon bond if market interest rate is 7.25% B. With a market interest rate of 5.5%, determine the loan amount for someone agreeing to make...
Let $N$ be a positive integer. Consider the relation $\circledast$ among pairs of integers $r, s \in \mathbb{Z}$ defined as $r \circledast s$ when $r-s$ is an integer multiple of $N$. Prove that...
Draw a circuit diagram for a typical home hair dryer. To which form (or forms) of energy is electric potential energy converted when you use the dryer?
Draw a vector field diagram for particles carrying charges $+2 q$ and $-q$ separated by a distance $r$. Comment on the significance of the vector diagram.
(a) Show that the Jones matrix of a polarization analyzer set at angle $\alpha$ to the $X$-axis is given by \[ \underline{\mathbf{L}}(\alpha)=\left[\begin{array}{cc} \cos ^{2} \alpha & \sin...
Let $\mathbf{V}(t)$ be a linearly filtered complex-valued, wide-sense stationary random process with sample functions given by \[ \mathbf{v}(t)=\int_{-\infty}^{\infty} \mathbf{h}(t-\tau)...
Santiago, a widower, made a will leaving $1 to his son, Carlos, and the balance in equal shares to his other children, Benito and Angelita. The estate, after deducting debts, taxes, and expenses,...
Refer to the information from Exercise 22-19. Use the information to determine the (1) Weighted average contribution margin , (2) Break-even point in units, and (3) Number of units of each product...
A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
An airplane flies from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x(t) be the...
The number N (in thousands) of cellular phone subscribers in Malaysia is shown in the table. (Midyear estimates are given.) (a) Use the data to sketch a rough graph of N as a function of (b) Use your...
Be prepared to explain the texts comprehensive To illustrate the issues related to interest capitalization, assume that on November 1, 2016, Shalla Company contracted Pfeifer Construction Co. to...
On April 1, 2020. Indigo Company received a condemnation award of $473,000 cash as compensation for the forced sale of the company's land and building, which stood in the path of a new state highway....
The market price of a stock is $24.55 and it is expected to pay a dividend of $1.44 next year. The required rate of return is 11.23%. What is the expected growth rate of the dividend? Submit Answer...

Let f: Rn R. Suppose that for each unit vector u Rn, the directional derivative

Question:

Step by Step Answer:

Let Ft fa tu and observe by definition that Thus F ...View the full answer

An Introduction to Analysis

Students also viewed these Numerical Analysis questions