6. Linear Approximation a. Suppose you have a function f(x), and suppose you know df|3 =...
6. Linear Approximation
a. Suppose you have a function f(x), and suppose you know df|3 = −4 dx. What is the equation of the tangent line to y = f(x) at x = 3, if f(3) = 7? And give an estimate of f(2.8).
b. The volume of a sphere of radius r is V = 1 3 πr3 . Find dV in terms of dr. Then find dV V in terms of dr r , and use it to answer the following question: If the radius is increased by 2%, by about how much is the volume increased?
c. Let f(x) = ln(x). Find the tangent line to the graph y = ln(x) at x = 1, and find df|1. Use it to estimate ln(1.03) and ln(0.98). By sketching a quick graph, say if your estimates are too high or too low
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6. Linear Approximation
a. Suppose you have a function f(x), and suppose you know df|3 =...
(3 points) The figure shows how a function f (x) and its linear approximation (.e., its...
(3 points) The figure shows how a function f (x) and its linear approximation (.e., its tangent line) change value when I changes from co to co + dr. y = f(x) fredr) Suppose f(x) = x2 + 2x, xo = 2 and dr = 0.05. Your answers below need to be very precise, so use many decimal places. (a) Find the change Af = f (30+ dc) - f(:30). Af Error = 14f-df Af = f(x + dr) -...
6. For a certain function f(x) we have: f'(x) = (x - 3)²(2x - 3) and...
6. For a certain function f(x) we have: f'(x) = (x - 3)²(2x - 3) and • f"(x) = 6(x - 3)(x - 2) (a) Use f' to find the intervals where f is increasing, the intervals where f is decreasing, the x- coordinates and nature (max, min or neither) of any local extreme values. (b) Use f" to find the intervals where the graph of f is concave up, the intervals where the graph of f is concave down...
2. Suppose the linear approximation of a differentiable function f(x, y, z) at the point (1,2,3)...
2. Suppose the linear approximation of a differentiable function f(x, y, z) at the point (1,2,3) is given by L(x, y, z) = 17+ 6(x – 1) – 4(y – 2) + 5(2 – 3). Suppose furthermore that x, y and z are functions of (s, t), with (x(0,0), y(0,0), z(0,0)) = (1, 2, 3), and the differentials computed at (s, t) = (0,0) are given by dx = 7ds + 10dt, dy = 4ds – 3dt, dz = 2ds...
Suppose that f(x) is a differentiable function such that the tangent line at x = 3...
Suppose that f(x) is a differentiable function such that the tangent line at x = 3 is given by y=-***. How many of the following statements MUST be true? I. According to the linearization of fat x = 3. f3.001) - 0.9989 IL (3) -0. III. f is concave down on an open interval containing x = 3. IV. The graph of y = f(x) attains a maximum value on the interval (-1,4). V. Applying Newton's Method to approximate the...
Question 2, non-calculator Question 1, calculator The curve C in the x-y-plane is given parametrically by (x(t), y(t), where dr = t sine) and dv = cos| t The Maclaruin series for a function f is give...
Question 2, non-calculator Question 1, calculator The curve C in the x-y-plane is given parametrically by (x(t), y(t), where dr = t sine) and dv = cos| t The Maclaruin series for a function f is given by r" for 1 sts 6 a) Use the ratio test to find the interval of convergence of the Maclaurin series for f a) Find the slope of the line tangent to the curve C at the point where t 3. b) Let...
For question 2 : Find the domain, y and x intercepts, f'(x), f''(x), Maxima and minna, the...
For question 2 : Find the domain, y and x intercepts, f'(x),
f''(x), Maxima and minna, the table for increasing and
decreasing, table for concavity and then sketching
2. Draw the graph of f(x) = x In(\xD) - (x – 4) In(x – 41). 3. Use cubic approximation to estimate the value of ln(1.3).
As we have seen, the total differential for a state function f (x, y) (an exact...
As we have seen, the total differential for a state function f (x, y) (an exact differential) can be written df =[∂f/∂x]y dx + [∂f/∂y]xdy The Euler criterion for the exactness of a differential states that the differential is exact if and only if df = M(x, y)dx + N(x, y)dy = ∂N [∂M/∂y]x = [∂N/∂x]y State whether the following differentials are exact or inexact. a) dq = CvdT + (RT/V) dV (assume that Cv and R are constants) b)...
5Suppose we have an objective function f(x,y) and a constraint y-h). Suppose the Lagrangian has a...
You can just answer question
bcd
5Suppose we have an objective function f(x,y) and a constraint y-h). Suppose the Lagrangian has a critical point at (0,0,X). Explain in a sentence or two how you know that line r(t) = (t,th,(0)) is tangent to the constraint. b At the critical point, compute the second derivative of f along the line in a d2 At the critical point, compute the second derivative of f along the graph y - h(x) Describe the...
1) 2) 3) Use linear approximation, i.e. the tangent line, to approximate 15.22 as follows: Let...
1)
2)
3)
Use linear approximation, i.e. the tangent line, to approximate 15.22 as follows: Let f(x) = z² and find the equation of the tangent line to f(x) at x = 15. Using this, find your approximation for 15.22 Given the function below f(x) = -180x3 + 396 1. Answer in mx + b form. Find the equation of the tangent line to the graph of the function at x = L(2) Use the tangent line to approximate f(1.1)....
2. Consider the function f : R2 → R2 given by. (x,y) (a) Compute the Df(x, y) (b) List every vect...
2. Consider the function f : R2 → R2 given by. (x,y) (a) Compute the Df(x, y) (b) List every vector r e R2 such that Df(ri, r2) 0. What can we say about the tangent plane to the surface of the graph at (ri,2,f(r1, r2))? (c) How do you know that the Hessian, Df(x, y) is necessarily symmetric? Recall that t,y D2 f(x,y) , y) (d) What are the eigenva of D2f(r1,r2) for each root of the gradient that...
(3 points) The figure shows how a function f (x) and its linear approximation (.e., its tangent line) change value when I changes from co to co + dr. y = f(x) fredr) Suppose f(x) = x2 + 2x, xo = 2 and dr = 0.05. Your answers below need to be very precise, so use many decimal places. (a) Find the change Af = f (30+ dc) - f(:30). Af Error = 14f-df Af = f(x + dr) -...
6. For a certain function f(x) we have: f'(x) = (x - 3)²(2x - 3) and • f"(x) = 6(x - 3)(x - 2) (a) Use f' to find the intervals where f is increasing, the intervals where f is decreasing, the x- coordinates and nature (max, min or neither) of any local extreme values. (b) Use f" to find the intervals where the graph of f is concave up, the intervals where the graph of f is concave down...
2. Suppose the linear approximation of a differentiable function f(x, y, z) at the point (1,2,3) is given by L(x, y, z) = 17+ 6(x – 1) – 4(y – 2) + 5(2 – 3). Suppose furthermore that x, y and z are functions of (s, t), with (x(0,0), y(0,0), z(0,0)) = (1, 2, 3), and the differentials computed at (s, t) = (0,0) are given by dx = 7ds + 10dt, dy = 4ds – 3dt, dz = 2ds...
Suppose that f(x) is a differentiable function such that the tangent line at x = 3 is given by y=-***. How many of the following statements MUST be true? I. According to the linearization of fat x = 3. f3.001) - 0.9989 IL (3) -0. III. f is concave down on an open interval containing x = 3. IV. The graph of y = f(x) attains a maximum value on the interval (-1,4). V. Applying Newton's Method to approximate the...
Question 2, non-calculator Question 1, calculator The curve C in the x-y-plane is given parametrically by (x(t), y(t), where dr = t sine) and dv = cos| t The Maclaruin series for a function f is given by r" for 1 sts 6 a) Use the ratio test to find the interval of convergence of the Maclaurin series for f a) Find the slope of the line tangent to the curve C at the point where t 3. b) Let...
For question 2 : Find the domain, y and x intercepts, f'(x),
f''(x), Maxima and minna, the table for increasing and
decreasing, table for concavity and then sketching
2. Draw the graph of f(x) = x In(\xD) - (x – 4) In(x – 41). 3. Use cubic approximation to estimate the value of ln(1.3).
You can just answer question
bcd
5Suppose we have an objective function f(x,y) and a constraint y-h). Suppose the Lagrangian has a critical point at (0,0,X). Explain in a sentence or two how you know that line r(t) = (t,th,(0)) is tangent to the constraint. b At the critical point, compute the second derivative of f along the line in a d2 At the critical point, compute the second derivative of f along the graph y - h(x) Describe the...
1)
2)
3)
Use linear approximation, i.e. the tangent line, to approximate 15.22 as follows: Let f(x) = z² and find the equation of the tangent line to f(x) at x = 15. Using this, find your approximation for 15.22 Given the function below f(x) = -180x3 + 396 1. Answer in mx + b form. Find the equation of the tangent line to the graph of the function at x = L(2) Use the tangent line to approximate f(1.1)....
2. Consider the function f : R2 → R2 given by. (x,y) (a) Compute the Df(x, y) (b) List every vector r e R2 such that Df(ri, r2) 0. What can we say about the tangent plane to the surface of the graph at (ri,2,f(r1, r2))? (c) How do you know that the Hessian, Df(x, y) is necessarily symmetric? Recall that t,y D2 f(x,y) , y) (d) What are the eigenva of D2f(r1,r2) for each root of the gradient that...
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