Question

4. The figure below shows the slope field for a differential equation

Answer 1

4. The figure below shows the slope field for a differential equation
\[\frac{ dy }{ dx }=f(x)\]
. Let
\[g(x) =\int\limits_{a}^{x}f(t)dt + C\]
be the family of functions that are solutions of the differential equation.

(a) Determine to the nearest integer the value of x for which all of the members of the family of g(x) will have a relative minimum value. Explain how you know.



(b) Determine to the nearest integer the value of x for which all of the members of the family of g(x) will have a relative maximum value. Explain how you know.



(c) On the figure below sketch the member of the family of g(x) for which g(0) = –3. (You may copy the figure on to a separate paper and fax to your instructor or you may scan it and attach it to your assignment)



(d) For the function sketched in part (c), determine the solution(s) of g(x)=0 to the nearest integer.

Answer 2

2nd fundamental theorem of calculus

Answer 3

oh shiznet, I just realized that this may be beyond calc2

Answer 4

nope, im in calculus AB

Answer 5

oh then this should be easy

Answer 6

keyword is should be, but im completely confused :/

Answer 7

FTC is definitely to be used for the first two parts...

Answer 8

By the FTC, you have that \(\dfrac{d}{dx}\displaystyle\int_a^x f(t)~dt=f(x)\), which is exactly \(\dfrac{dy}{dx}\), as given. Apparently, there's a graph you can refer to to answer the first two questions.

If in the figure, \(f(x)\) is negative for some interval \((a,b)\), then positive for some interval \((b,c)\), then \(g(x)\) attains a minimum at \(x=b\).

If \(f(x)\) is positive for some interval \((a,b)\), then negative for some interval \((b,c)\), then \(g(x)\) attains a max at \(x=b\).

Answer 9

Ah I see you're provided a slope field. I'm sure you understand that \(\dfrac{dy}{dx}\) is negative if you have a downward slope, \, and positive if upward, /

Answer 10

is the keyword here relative minimum @SithsAndGiggles ?

Answer 11

Yep, I would take that to mean you're intended to use the derivative test to find extrema.

Answer 12

yes