4. Consider functions
4. Consider functions \(f(x, y)=\sqrt[3]{x^{3}+y^{3}}\) and \(\mathbf{c}(t)=(t, 2 t)\).
(a) Show that \(f_{x}(0,0)\) and \(f_{y}(0,0)\) exist. What is \(\nabla f(0,0)\) ?
(b) Show that \(f\) is not differentiable at the point \((0,0)\).
(c) Find \((f \circ c)(t)\) and then compute its derivative at the point \(t=0\).
(d) Show that \((f \circ \mathbf{c})^{\prime}(0) \neq \nabla f(\mathbf{c}(0)) \cdot \mathbf{c}^{\prime}(0)\). Does this contradict the chain rule formula? why? why not?
Homework Answers
Question 1. Compute the derivative of the following functions.
Question 1. Compute the derivative of the following functions.(a) \(f(x)=x^{3}-\frac{2}{\sqrt{x}}+4\)(b) \(f(x)=2^{3 x-1}\)(c) \(f(x)=\ln \left(5 x^{2}+1\right)\)(d) \(f(x)=\frac{\tan (x)}{x^{2}+1}\)(e) \(f(x)=e^{x^{2}} \cdot \arctan (2 x)\)(f) \(f(x)=\sin (x)^{2} \cdot\left(\tan (x)+\cos (x)^{2}\right)\).Question 2. In geometry, the folium of Descartes is a curve given by the equation$$ x^{3}+y^{3}-3 a x y=0 $$Here, \(a\) is a constant.The curve was first proposed by Descartes in 1638 . Its claim to fame lies in an incident in the development of calculus. Descartes challenged Fermat to find the tangent line...
Consider the function
Consider the function \(f(x, y)=\frac{x y}{x^{2}+y^{2}}\) if \((x, y) \neq(0,0)\)$$ =0 \text { if }(x, y)=(0,0) $$Which one of the statement is incorrect.Select one:a. \(f(x, y)\) is differentiable everywhere.b. \(f(x, y)\) is differentiable everywhere except at the origin.c. \(f(x, y)\) is not continuousd. First partial derivatives \(f(x, y)\) exist.e. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{x^{2}+y^{2}}\) does not exist.
showing multivariable calculus functions are differentiable Please help! 2. Recall that by Theorem 3 of Section 14.3, a...
showing multivariable calculus functions are differentiable
Please help!
2. Recall that by Theorem 3 of Section 14.3, a function f(x,y) is differentiable if its partial derivatives fa and fy both exist and are continuous. (a) Use this idea to show that the function f(x,y)-esin ry is differentiable. (b) Let o be a differentiable function and f(,)Jy Find the partial derivatives of f and determine whether they are continuous. Hint: The Fundamental Theorem of Calculus gives us that Ø has an...
Suppose that the functions
Suppose that the functions \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}, g: \mathbb{R}^{3} \rightarrow \mathbb{R}\), and \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}\) are continuously differentiable and let \(\left(x_{0}, y_{0}, z_{0}\right)\) be a point in \(\mathbb{R}^{3}\) at which$$ f\left(x_{0}, y_{0}, z_{0}\right)=g\left(x_{0}, y_{0}, z_{0}\right)=h\left(x_{0}, y_{0}, z_{0}\right)=0 $$and$$ \left\langle\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right) \times \nabla h\left(x_{0}, y_{0}, z_{0}\right)\right\rangle \neq 0 $$By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a...
Complex analysis
Complex analysis(i) If \(f\) is differentiable at \(z_{0}\) then \(f\) is continuous at \(z_{0}\).(ii) If \(f\) and \(g\) are differentiable at \(z_{0}\), then \(f+g\) and \(f g\) also are, and \((f+g)^{\prime}\left(z_{0}\right)=f^{\prime}\left(z_{0}\right)+g^{\prime}\left(z_{0}\right) \quad\) (sum rule); \((f g)^{\prime}\left(z_{0}\right)=f^{\prime}\left(z_{0}\right) g\left(z_{0}\right)+f\left(z_{0}\right) g^{\prime}\left(z_{0}\right) \quad\) (product rule). If in addition \(g\left(z_{0}\right) \neq 0\), then \(f / g\) is differentiable at \(z_{0}\), and \(\left(\frac{f}{g}\right)^{\prime}\left(z_{0}\right)=\frac{f^{\prime}\left(z_{0}\right) g\left(z_{0}\right)-f\left(z_{0}\right) g^{\prime}\left(z_{0}\right)}{g\left(z_{0}\right)^{2}} \quad\) (quotient rule).(iii) If \(f\) is differentiable at \(z_{0}\) and \(g\) is differentiable at \(f\left(z_{0}\right)\), then the composite function \(g \circ...
Please help out is somewhat difficult. In practice, it is often easier to show a stronger...
Please help out
is somewhat difficult. In practice, it is often easier to show a stronger condition: if each partial derivative OJi = 1, ... , n, is continuous in a disc around p = _ (a1.... , an), then f is differentiable дх, (a1,., an) Put differently: if f is continuously differentiable at p, it is differentiable at However, just as in the one-variable case, there are functions that are differentiable but not at p = p. continuously differentiable....
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x +...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
An object moves though a vector field.
1. ( 8 points) An object moves though a vector field, \(\overrightarrow{\mathbf{F}}(x, y)\), along a circular path, \(\overrightarrow{\mathbf{r}}(t)\), starting at \(P\) and ending at \(Q\) as shown in the graph below.(a) At the point \(R\) draw and label a tangent vector in the direction of \(d \overrightarrow{\mathbf{r}}\).(b) At the point \(R\) draw and label a vector in the direction of the vector filed, \(\overrightarrow{\mathbf{F}}(R)\).(c) At the point \(R\) is \(\overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}}\) positive, negative, or zero? Circle the correct...
Exercise 1. Do the following: (a) Write a statement defining the Chain Rule for the functions...
Exercise 1. Do the following: (a) Write a statement defining the Chain Rule for the functions g: R" → Rm and f: RM + RP. Then describe how it works in a paragraph, assuming the reader is a classmate who has been following the course but missed the lecture on Properties of the Derivative. (b) Explain in detail how the Chain Rule you learned in Calculus I, (fog)(x) = f (g(2)).g'(x), is really just the special case of your statement...
showing multivariable calculus functions are differentiable
Please help!
2. Recall that by Theorem 3 of Section 14.3, a function f(x,y) is differentiable if its partial derivatives fa and fy both exist and are continuous. (a) Use this idea to show that the function f(x,y)-esin ry is differentiable. (b) Let o be a differentiable function and f(,)Jy Find the partial derivatives of f and determine whether they are continuous. Hint: The Fundamental Theorem of Calculus gives us that Ø has an...
Please help out
is somewhat difficult. In practice, it is often easier to show a stronger condition: if each partial derivative OJi = 1, ... , n, is continuous in a disc around p = _ (a1.... , an), then f is differentiable дх, (a1,., an) Put differently: if f is continuously differentiable at p, it is differentiable at However, just as in the one-variable case, there are functions that are differentiable but not at p = p. continuously differentiable....
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
1. ( 8 points) An object moves though a vector field, \(\overrightarrow{\mathbf{F}}(x, y)\), along a circular path, \(\overrightarrow{\mathbf{r}}(t)\), starting at \(P\) and ending at \(Q\) as shown in the graph below.(a) At the point \(R\) draw and label a tangent vector in the direction of \(d \overrightarrow{\mathbf{r}}\).(b) At the point \(R\) draw and label a vector in the direction of the vector filed, \(\overrightarrow{\mathbf{F}}(R)\).(c) At the point \(R\) is \(\overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}}\) positive, negative, or zero? Circle the correct...
Exercise 1. Do the following: (a) Write a statement defining the Chain Rule for the functions g: R" → Rm and f: RM + RP. Then describe how it works in a paragraph, assuming the reader is a classmate who has been following the course but missed the lecture on Properties of the Derivative. (b) Explain in detail how the Chain Rule you learned in Calculus I, (fog)(x) = f (g(2)).g'(x), is really just the special case of your statement...
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