Math

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ttp://calculus.sfsu.edu/latexrender/pictures/e454953d0aa505edd5ae923bbc6a28b6.png

Lots of clips on stream. Try 'Fundamental Theorem' as the keyword.

1. Graph (t) = 5. Mark = 1and = xon the http://calculus.sfsu.edu/latexrender?src=%5Cbegin%7Bdisplaymath%7D+%5Ctextstyle%7Bt%7D+%5Cend%7Bdisplaymath%7D&type=png&size=15-axis. Using the formula for the area of a rectangle, find the function: (x) = integral from 1^x 5 dt. Then calculate '(x)?

2. Graph (t) = 3-3 t. Assume that 1 < x < 1and using the formula for the area of triangles (or trapezoids) find the function: (x) = integral from (-1)^x(3-3 t) dt. Then calculate '(x).

3. Based on the graph (t)sketched to the left, (a) fill out approximate values in the table of (x) = integral from 0^x f(t) dt. (b) For what value of http://calculus.sfsu.edu/latexrender?src=%5Cbegin%7Bdisplaymath%7D+%5Ctextstyle%7Bx%7D+%5Cend%7Bdisplaymath%7D&type=png&size=15in the interval 0,4]does (x)reach a minimum? (Hint 1. you see the graph of (t)not (x). Hint 2. (2)is not the minimum. )

(c) Sketch an approximate graph of the area function (x)restricted to the interval 0,4].

ttp://calculus.sfsu.edu/latexrender/pictures/9f3897dace48d8a263a2e8e6513105ab.png

4. Use the Fundmamental Theorem of Calculus to find /dx integral from 2^x cos (t^4) dt.

5. Find /dx integral from x^6 cos (root (s^4+1)) ds.

6. Find /dx integral from 2^(3 x+1) ln (t+1) dt.

7. Find /dx integral from (-x)^x[z-1]/[z+2] dz.

8. Find /dx integral from (-3x)^(2 x) e^(t^2) dt.

9. (ask, discuss - and share your ideas, but no need to submit) The graph of a function http://calculus.sfsu.edu/latexrender?src=%5Cbegin%7Bdisplaymath%7D+%5Ctextstyle%7Bf%7D+%5Cend%7Bdisplaymath%7D&type=png&size=15is sketched below. Sketch the antiderivative (x)of http://calculus.sfsu.edu/latexrender?src=%5Cbegin%7Bdisplaymath%7D+%5Ctextstyle%7Bf%7D+%5Cend%7Bdisplaymath%7D&type=png&size=15defined on the interval 0,3]and such that (0) = 0.

Hint: What does (x)represent on the graph of (t)?

ttp://calculus.sfsu.edu/latexrender/pictures/526f5b02f044d53943ef29f68daeae2d.png

ttp://calculus.sfsu.edu/latexrender/pictures/05a3ab980b02c755dc4334fb272e81ef.png

10. Please discuss but do not submit. Share your favorite example of a function for which the related area so far function has a nice and intuitive interpreation in real life. Maybe something like the can example from the lecture.

11. Discuss it please but do not submit. Your participation in this discussion will help you understand the first part of the next lesson.

(a) Sketch a graph of (x) = integral from 1^x t^2 dtby first sketching a graph of ^2and by counting small squares. (b) Note that (1) = 0(Why?). (c) Also, note that the FTC gives us '(x) = x^2.Why? Based on this information how can we obtain an algebraic formula for (x)?