Slide 2:

1.a) - we filled in the chart by calculating the integrals (signed areas) under the graph, start from in this case, -2 because the integral is from -2 to x.

1.b) - we sketched the graph simply by plotting the points we gathered from our table in question a.

1.c) - we analyzed the graph for local extrema

- the only critical number is x=2 because at that value of x, a local minimum is present

- end points are not critical numbers because there is no way of distinguishing if it is a local maximum or minimum without checking the slope on both sides of the point

1.d) -we simply found where the graph was increasing by looking for a positive slope

Slide 3:

a, b.) - to obtain these solutions, we can simply replace the variable t with x, included in the domain of the integral, due to the fundamental theorem of calculus

brief description of process:

- F(X) = f(b) - f(a)

- from this we get: (-cos(x^3)) - (-cos(pi))

- now we want to find the derivative of this integral (stated in the quesiton)

- we get: sin(x^3)

- the derivative of a constant is 0

c.) -the solution to this question was similar, however, since the integral is from x to 1, we can make the integral NEGATIVE to make the integral from 1 to x instead

Slide 4:

d.) - the solution is similar to that of question c on slide 2

e, f.) - these questions include a slight different solution

- since they are an accumulation of functions, to find the derivative, we must apply the chain rule: (f'(g(x)) (g'(x))

Slide 5:

- to find the amount of gallons using the rate (derivative), we integrate it from 0 to 4 hours

- yes, it's that simple =)

Slide 6:

- the only trick to this question is the +3 included in the integral

- we solve this part by find the integral of 3, from 4 to 7

Slide 7:

- points of intersections are found by making the two functions equal each other

- then, but inputting values between the pair of intersections (-2, 0) & (0, 3) we find which graph is on top of the other, in each case (in this case, twice)

Slide 8:

- we applied: (integral from (-2) - 0) ((top function) - (bottom function)) + (integral from 0 - 3) ((top function) - (bottom function))

- solve algebraically, applying rules of finding the derivative of an integral

- grunttt workkk

THE END

Okay I hope you guys found this slightly helpful. It wasn't as specific as it should be but yeahh. Tomorrow's scribe is.. Etimz since he was the only one who hasn't done his fifth scribe yet.. or something like that.

Good luck on the pre-test tomorrow! bye =)

Slide 5:

- to find the amount of gallons using the rate (derivative), we integrate it from 0 to 4 hours

- yes, it's that simple =)

Slide 6:

- the only trick to this question is the +3 included in the integral

- we solve this part by find the integral of 3, from 4 to 7

Slide 7:

- points of intersections are found by making the two functions equal each other

- then, but inputting values between the pair of intersections (-2, 0) & (0, 3) we find which graph is on top of the other, in each case (in this case, twice)

Slide 8:

- we applied: (integral from (-2) - 0) ((top function) - (bottom function)) + (integral from 0 - 3) ((top function) - (bottom function))

- solve algebraically, applying rules of finding the derivative of an integral

- grunttt workkk

THE END

Okay I hope you guys found this slightly helpful. It wasn't as specific as it should be but yeahh. Tomorrow's scribe is.. Etimz since he was the only one who hasn't done his fifth scribe yet.. or something like that.

Good luck on the pre-test tomorrow! bye =)

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