How to find:

asked 2021-05-14

Use the given graph off over the interval (0, 6) to find the following.

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)

b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)

c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)

d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)

e) The coordinates of the point of inflection. \((x,\ y)=\)

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)

b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)

c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)

d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)

e) The coordinates of the point of inflection. \((x,\ y)=\)

asked 2021-05-03

Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}.x=e^t,y=te^{-t}\). For which values of t is the curve concave upward?

asked 2021-09-11

Find \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}\) and \(\displaystyle{\frac{{{d}^{{2}}{y}}}{{{\left.{d}{x}\right.}^{{2}}}}}.{x}={e}^{{t}},{y}={t}{e}^{{-{t}}}\). For which values of t is the curve concave upward?

asked 2021-05-16

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.

A. Let y=f(x) be the equation of C. Find f(x).

B. Find the slope at P of the tangent to C.

C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?

D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.

E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.

Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

A. Let y=f(x) be the equation of C. Find f(x).

B. Find the slope at P of the tangent to C.

C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?

D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.

E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.

Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

asked 2021-11-21

\(x = t^2+ 1,\ y = t^2 + t\)

asked 2021-06-01

Use Part 2 of the fundamental Theorem of Calculus to find the derivatives.

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{0}}}^{{{x}}}}{e}^{{\sqrt{{{t}}}}}{\left.{d}{t}\right.}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{0}}}^{{{x}}}}{e}^{{\sqrt{{{t}}}}}{\left.{d}{t}\right.}\)