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Question

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 2021-05-30

Let \(g(x) = \int_0^x f(t) dt\) where f is the function whose graph is shown in the figure.

(a) Estimate g(0), g(2), g(4), g(6), and g(8).

(b) Find the largest open interval on which g is increasing. Find the largest open interval on which g is decreasing.

(c) Identify any extrema of g.

(d) Sketch a rough graph of g.

(a) Estimate g(0), g(2), g(4), g(6), and g(8).

(b) Find the largest open interval on which g is increasing. Find the largest open interval on which g is decreasing.

(c) Identify any extrema of g.

(d) Sketch a rough graph of g.

asked 2021-05-26

If f is continious on, what you can say about it's graph?

asked 2021-05-29

Let f and g be differentiable functions. Find a formula for \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\frac{{{3}{f{{\left({x}\right)}}}}}{{{2}}}}-{5}{g{{\left({x}\right)}}}\right]}\)