Show all work if full or partial credit is desired. No calculator on this first part, and no notes, books, or websites allowed.

Part 1: No Calculator

State an antiderivative for each functions below:

1. (2 points)   `int 3 dx =`
2. (2 points)   `int (3x+1) dx =`
3. (2 points)   `int x^2 dx =`
4. (2 points)   `int cos(x) dx =`
5. (2 points)   `int e^(3x) dx =`
6. (2 points)   `int 1/x dx =`
7. (2 points)   `int sec^2(x) dx =`
8. (2 points)   `int 1/sqrt(1-x^2) dx =`.

9. (10 points)   State the Fundamental Theorem of Calculus part 1.

Show all work if full or partial credit is desired. You may use a graphing calculator or Desmos or Geogebra. If Geogebra is used, then the device must be in airplane-mode (i.e., Internet, cellular, or Bluetooth connections). No notes, books or websites allowed. No derivative nor integral calculators allowed.

Part 2: Calculator Allowed

Work must be shown.

10. (10 points)   Find two (2) of the indefinite integrals below.
    1. `int_0^4 (1/2 x + 1) dx=`
    2. `int_0^4 (4x-x^2) dx=`
    3. `int_0^pi sin(x) dx=`
    4. `int_1^5 1/x dx=`
    5. `int_((-5pi)/6)^((5pi)/6) sin(x) dx=`
    6. `int_0^(pi/2) cos^3(x) sin(x) dx=`
11. (5 points)   Graph one of the definite integrals above, shading the the area(s) found.

12. (5 points)   Find two (2) of the indefinite integrals below.
    1. `int sec(x) tan(x) dx=`
    2. `int tan(x) dx=`
    3. `int (x cos(x^2))dx=`
    4. `int (x^2+5x+7)/(x+3) dx=`
    5. `int 1/(sqrt(4-x^2)) dx=`
    6. `int 1/(1+x^2) dx=`

Fun with Integrals

13. (15 points)   Complete the following for `f(x)=3x+1`.
    1. Graph `f(x)` on the interval `[-1,3]`.
    2. Estimate the area of `f(x)` for `x` in `[0,2]` by partitioning the interval into `n=4` subintervals, using right end points. Draw this.
    3. State `Delta x` and `x_i` for the interval `[0,2]` with `n` subintervals and right endpoints.
    4. State the Riemann sum for `f(x)` on `[0,2]` with `n` subintervals and right endpoints.
    5. Simplify the Riemann sum using necessary sum formulas. The formulas listed below might help.
       • `sum_{i=1}^n i = (n(n+1))/2`
       • `sum_{i=1}^n i^2 = (n(n+1)(2n+1))/6`
       • `sum_{i=1}^n i^3 = (n^2(n+1)^2)/4`
    6. Find the limit as `n` increases of the Riemann sum that you found above.
    7. State the net area of `f(x)` over the interval `[0,2]`.

Fun with Integrals

14. (10 points)   `p(x)=x^3-2x^2-x+2`
    1. Graph `p(x)` so you can clearly see it on the interval `[-1,3]`.
    2. What is the net area of `p(x)` over `[-1,3]`?
    3. What is the total area of `p(x)` over `[-1,3]`?
15. (10 points)   A ball is thrown into the air at 20 meters per second. Its velocity `t` seconds later is given by `v(t)=−9.8t+20`.
    1. What is the net change in distance from `t=0` to `2` seconds?
    2. What is the net change in distance from `t=0` to `4` seconds?
    3. What is the total distance traveled from `t=0` to `4` seconds?

Apply to Fundamental Theorem of Calculus part 1 to the following problems

16. (5 points)   `d/dx [int_1^x sin(t^2)dt]`
17. (5 points)   `d/dx [int_1^(x^2) sin(t)dt]`