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2. Problem 2 Let g(z) be a differentiable function defined on is shown below. Also suppose that g...
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so...
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so that the function h()f(g(a))g'(a) is integrable with anti-derivative H. Prove that f is integrable, and that nog . Is an antiderivative for J.
Consider the function below. y z= 2+x+ (a) Match the function with its graph (labeled A-F)....
Consider the function below. y z= 2+x+ (a) Match the function with its graph (labeled A-F). (b) Match the function with its contour map (labeled 1-VI). O V VI Give reasons for your choices. Select- Also, the values of z approach 0 as we use This function is not periodic, ruling out the graphs in (b) Match the function with its contour map (labeled I-VI) II III IV V VI Give reasons for your choices. This function is not periodic,...
(a) A function / has first derivative f'(z) = and second derivative 3) f"(x) It is also known that the function...
(a) A function / has first derivative f'(z) = and second derivative 3) f"(x) It is also known that the function f has r-intercept at (-3,0), and a y-intercept at (0,0) (i) Find all critical points, and use them to identify the intervals over which you will examine the behaviour of the first derivative ii) Use the f'(), and the First Derivative Test to classify each critical point. (ii) Use the second derivative to examine the concavity around critical points...
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...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
4) Three friends are working on a problem concerning a differentiable function f(x,y,z) at a point...
4) Three friends are working on a problem concerning a differentiable function f(x,y,z) at a point P. While these 3 people are discussing their calculations: 1st person says: "I found the equation of the tangent plane to the surface f(x, y, z)=c at the point P to be 3x + 6y – 2z =9". 2nd person says: "I discovered that the maximum value of the directional derivative of the function f at the point P is 5" 3rd person says:...
7.7.4 The hypotheses of Theorem 7.24 require that f be differentiable on all of the interval I. Y...
7.7.4 The hypotheses of Theorem 7.24 require that f be differentiable on all of the interval I. You might think that a positive derivative at a single point also implies that the function is increasing, at least in a neighborhood of that point. This is not true. Consider the function /(z) _{0,/2 + ra sin.ri. if 0 (e) Prove that if a function F is differentiable on a neighborhood of ro with F(ro)0 and F is continuous at zo, then...
2. Find the intervals on which the following function is continuous. tan r B( ) = V4-12 3. Find the derivatives o...
2. Find the intervals on which the following function is continuous. tan r B( ) = V4-12 3. Find the derivatives of the following functions (a) f(z)= /5x +1 (b) gla) = (2r -3)(a2 +2) (c) y = In (x + Vx2 - 1) 4. Find the intervals on which the following function is decreasing. f(z) = 36x +3r2 - 2r 5. Evaluate the following integrals. r dr (a) sec2 tan (b) dar 3 (c) da 5x+1 1. Sketch the...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
Compute the partial derivatives of the function at the given point, and determine what the total...
Compute the partial derivatives of the function at the given point, and determine what the total derivative would be if the function was differentiable there; then determine whether it is differentiable or not by definition; that is, determine whether or not 0. 22+y2 (x,y) # (0,0) fão + n) – $(20) - Df#, (h) lim ho hi 4. h(x, y) = 0 (2,y) = (0,0) at (0,0); 5. g(x, y, z) = x + 3y - 2 - 1 at...
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so that the function h()f(g(a))g'(a) is integrable with anti-derivative H. Prove that f is integrable, and that nog . Is an antiderivative for J.
Consider the function below. y z= 2+x+ (a) Match the function with its graph (labeled A-F). (b) Match the function with its contour map (labeled 1-VI). O V VI Give reasons for your choices. Select- Also, the values of z approach 0 as we use This function is not periodic, ruling out the graphs in (b) Match the function with its contour map (labeled I-VI) II III IV V VI Give reasons for your choices. This function is not periodic,...
(a) A function / has first derivative f'(z) = and second derivative 3) f"(x) It is also known that the function f has r-intercept at (-3,0), and a y-intercept at (0,0) (i) Find all critical points, and use them to identify the intervals over which you will examine the behaviour of the first derivative ii) Use the f'(), and the First Derivative Test to classify each critical point. (ii) Use the second derivative to examine the concavity around critical points...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
4) Three friends are working on a problem concerning a differentiable function f(x,y,z) at a point P. While these 3 people are discussing their calculations: 1st person says: "I found the equation of the tangent plane to the surface f(x, y, z)=c at the point P to be 3x + 6y – 2z =9". 2nd person says: "I discovered that the maximum value of the directional derivative of the function f at the point P is 5" 3rd person says:...
7.7.4 The hypotheses of Theorem 7.24 require that f be differentiable on all of the interval I. You might think that a positive derivative at a single point also implies that the function is increasing, at least in a neighborhood of that point. This is not true. Consider the function /(z) _{0,/2 + ra sin.ri. if 0 (e) Prove that if a function F is differentiable on a neighborhood of ro with F(ro)0 and F is continuous at zo, then...
2. Find the intervals on which the following function is continuous. tan r B( ) = V4-12 3. Find the derivatives of the following functions (a) f(z)= /5x +1 (b) gla) = (2r -3)(a2 +2) (c) y = In (x + Vx2 - 1) 4. Find the intervals on which the following function is decreasing. f(z) = 36x +3r2 - 2r 5. Evaluate the following integrals. r dr (a) sec2 tan (b) dar 3 (c) da 5x+1 1. Sketch the...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
Compute the partial derivatives of the function at the given point, and determine what the total derivative would be if the function was differentiable there; then determine whether it is differentiable or not by definition; that is, determine whether or not 0. 22+y2 (x,y) # (0,0) fão + n) – $(20) - Df#, (h) lim ho hi 4. h(x, y) = 0 (2,y) = (0,0) at (0,0); 5. g(x, y, z) = x + 3y - 2 - 1 at...
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