Question

In calculus the curvature of a curve that is defined by a function

y = f(x)

is defined as

κ =

y'' |

[1 + (y')^{2}]^{3/2} |

.

Find

y = f(x)

for which

κ = 1.

Answer #1

Let y = x 2 + 3 be a curve in the plane.
(a) Give a vector-valued function ~r(t) for the curve y = x 2 +
3.
(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint:
do not try to find the entire function for κ and then plug in t =
0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)|
dT~ dt (0) .]
(c) Find the center and...

The curvature at a point P of a curve y =
f(x) is given by the formula below.
k =
|d2y/dx2|
1 + (dy/dx)2
3/2
(a) Use the formula to find the curvature of the parabola
y = x2
at the point
(−2, 4).
(b) At what point does this parabola have maximum curvature?

The function f(x, y) is defined by
f(x, y) = 5x^3 * cos(y^3).
You will compute the volume of the 3D body below z = f(x, y) and
above the x, y-plane, when x
and y are bounded by the region defined between y = 2 and y =1/4 *
x^2.
(a) First explain which integration order is the preferred one in
this case and explain why.
(b) Then compute the volume.

What can you say about the curvature of the curve y = f(x) at
the point (x0, y0) and the
curvature of y = f −1(x) at the point (y0, x0)? Prove your
answer

] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

Let F be the defined by the function F(x, y) = 3 + xy - x - 2y,
with (x, y) in the segment L of vertices A (5,0) and B (1,4). Find
the absolute maximums and minimums.

6. Consider the function f defined by f (x, y) = ln(x
− y). (a) Determine the natural domain of f. (b) Sketch the level
curves of f for the values k = −2, 0, 2. (c) Find the gradient of f
at the point (2,1), that is ∇f(2,1). (d) In which unit vector
direction, at the point (2,1), is the directional derivative of f
the smallest and what is the directional derivative in that
direction?

1. Let f be the function defined by f(x) = x
2 on the positive real numbers. Find the
equation of the line tangent to the graph of f at the point (3,
9).
2. Graph the reflection of the graph of f and the line tangent to
the graph of f at the point
(3, 9) about the line y = x.
I really need help on number 2!!!! It's urgent!

If f:X→Y is a function and A⊆X, then define
f(A) ={y∈Y:f(a) =y for some a∈A}.
(a) If f:R→R is defined by f(x) =x^2, then find f({1,3,5}).
(b) If g:R→R is defined by g(x) = 2x+ 1, then find g(N).
(c) Suppose f:X→Y is a function. Prove that for all B,
C⊆X,f(B∩C)⊆f(B)∩f(C). Then DISPROVE that for all B, C⊆X,f(B∩C)
=f(B)∩f(C).

Produce graphs of f that reveal all the important aspects of the
curve. Then use calculus to find the following. (Enter your answers
using interval notation. Round your answers to two decimal
places.)
f(x) = 4 sin(x) + cot(x), −π ≤ x ≤ π
Find the interval of increase.
Find the interval of decrease.
Find the inflection points of the function.
(x, y) = (smaller x-value)
(x, y) = (larger x-value)
Find the interval where the function is concave up....

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