If I have the following function f(x,y) = x^3 + x^2y − 16y
There are 6 things I would like to know how to do:
I would like to know how to find the first-order and second-order partial derivatives of f?
If the value of f when x = 3 and y = −2 is 41 but I was to know only that the value of x lies between 2.98 and 3.02, while the value of y lies between −2.03 and −1.97, how would I use the first-order partial derivatives to investigate the accuracy of 41 as an estimate of f(x,y) for values of x and y in these ranges?
How would I use the chain rule to determine, in terms of t, the rate of change of f with time along the path given at time t by x(t) = 4 sec(t), y(t) = 1 − t^2?
If I have a surface where surface z = f(x,y) where f(x,y) is defined above how would I determine the gradient function grad f(x,y) and hence find the slope of the surface at the point (3, −2,41) in the direction of the unit vector j?
How would I determine the second-order Taylor polynomial for f about (−1,2)?
And finally how would I locate and classify all the stationary points of f?
Solution Preview
If I have the following function f(x,y) = x3 + x2y − 16y
There are 6 things I would like to know how to do:
I would like to know how to find the first-order and second-order partial derivatives of f?
If the value of f when x = 3 and y = −2 is 41 but I was to know only that the value of x lies between 2.98 and 3.02, while the value of y lies between −2.03 and −1.97, how would I use the first-order partial derivatives to investigate the accuracy of 41 as an estimate of f(x,y) for values of x and y in these ranges?
Solution. As f(x,y) = x3 + x2y − 16y, we have
and
So, and
Note that dx=0.02 and dy=0.03. Hence,
So, when the value of x lies between 2.98 and 3.02, ...
Solution Summary
The following posting helps with problems involving second-order Taylor polynomials.
Use Taylorpolynomials about 0 to evaluate sin(0.3) to 4dp,showing all workings.
1)F(x)=square root 4+x and G(x)=square root 1+x
by writing square root of 4+x=2 square root 1+1/4x
and using substitution in one of the standard Taylor series, find the Taylor series about 0 for f.Given explicitly all terms up to term in x raise
Rules and Applications of the Derivative
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1. Use the Product Rule to find the derivatives of the following functions:
a. f(X) = (1- X^2)*(1+100X)
b. f(X) = (5X + X^-1)*(3X + X^2)
c. f(X) = (X^.5)*(1-X)
d. f(X) = (X^3 + X^4)*(30
1. Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find the sum, then do so {see attachment}
2. Find the open interval of convergence and test the endpoints for absolute and conditional convergence: {see attachment}
3. For the equation f (x) = ... {see attachment
Determine whether the polynomials have multiple roots.
See attached file for full problem description.
19. Let F be a field and let f(x) =...... The derivative, D(f(x)), of f(x) is defined by
D(f(x)) = ......
where, as usual, ....... (n times). Note that D(f(x)) is again a polynomial with coefficients in F.
The polynomi
1 Write the Taylor polynomial with center zero and degree 4 for the function f(x) = e^-x
2 Determine the values of p for which the series
∞
Σ 1/(2p)ⁿ
n=1
3 Calculate the sum of the first ten terms of the series, then estimate the error
A). Prove that the function f(x) = e^x is differentiable on R, and that (e^x)' = e^x. ( Hint: Use the definition of e^x, and consider the sequence of partial sums.)
My thoughts on a:
I tried to prove the differentiability by proving continuity on R, since e^x is series, sum of polynomials, and all polynomials are different
