MATH 115 Final: MATH 115 UPenn 07aM115Final

• School: UPenn
• Department: Mathematics
• Course: MATH 115
• Professor: All

Document Summary

If we call the numbers x and y, then we have xy = 10. The derivative of this is 1 10/x2, which is equal to zero when x2 = 10, so x = 10. We need x to be positive, so x = 10, meaning the minimum sum is 10 + 10/ 10 = 2 10. 1: (14 points) evaluate the following integrals: (a) (b) We can use the substitution u = x3, which gives du = 3x2dx, so the integral becomes. We can use the substitution u = 1 + ex, which gives du = exdx, so the integral becomes. Z cos(u)du = sin(u) = sin(1 + ex) + c. Swap the limits so we can apply the rst part of the fundamental theorem of calculus, then it follows directly from ftc1. d dx. Z x dt ln t d dx x. Make the substitution u = 2 + sin(x), which gives du = cos(x)dx.