On the calculus exercises worksheet, please utilize the general forms of derivatives and integrals from the document in your packet (also found HERE) On the derivatives worksheet you are to take the first and second derivatives of each function, however, at present, for our needs you may SKIP questions 5, 6, 7, and you should consider #10 to be your challenge problem. Solutions for the first derivatives are below. If you can take a first derivative, taking a second follows the same pattern, at least for simple polynomials.
For the integrals/anti-derivatives, you can skip items 4, 5, and 9. Consider number 6 to be your challenge problem. For info on how to work out the “definite” integrals on 7 and 8, see the end of the power point notes HERE We will begin class on Wednesday with definite integrals just before we have our quiz. Give these a try, but don’t freak out if you don’t get them.
Hopefully these will help you out. If you are having difficulty with specific problems PLEASE email me or come in on Tuesday morning and let’s get it sorted out. Also, I kind of rushed these so if you find something amiss please let me know ASAP.
1) 21x^2 2) 6x+6 3) 30x^5-21x^2+3 4) -21x^6 8. re-write the function as x^-6, dy/dx = -6x^-7
9) let the function be written as 3x^(1/2), dy/dx = (3/2)x^(-1/2)
1) (4/2)x^2 + C = 2x^2 + C 2) x^3-3x^2+2x + C 3) x^4+(7/2)x^2 + C
7) anti-derivative = (4/3)x^3 + (3/2)x^2, now evaluate the anti-derivative at each limit and take the difference (top limit evaluation minus bottom), so:
(4/3)6^3 + (3/2)6^2 – 0 – 0
288 + 54 + 0 + 0
8. anti-derivative = ln x, now evaluate the anti-derivative at each limit and take the difference (top limit evaluation minus bottom), so:
ln5 – ln2 = ln5/ln2 = 2.32