MATH 100 V1A October 17th – Practice problems Hints and Solutions 1. In an ideal environment, a cell culture grows at a rate proportional to the number of cells present. Suppose a culture has 500 cells initially and 900 cells after 24 hours. How many cells will there be after an additional 10 hours? Solution: Let P (t) denote the number of cells in the culture at time t. We are told that the cell culture grows at a rate proportional to the number of cells present, i.e. dP = kP, dt for some constant k (to be determined). From class, we know that the solutions to this differential equation are of the form P (t) = Aekt , for some other constant A (also to be determined). We are also told that the cell culture initially has 500 cells, so 500 = P (0) = Ae0 = A. Therefore our equation simplifies to P (t) = 500ekt . Since there are 900 cells after 24 hours, we must have that 900 = P (24) = e24k . Solving this log( 9 ) for k gives us that k = 245 . So the equation that describes the number of cells in the culture as a function of time is given by log P (t) = 500e 9 t 5 . Now we can use this to solve the problem: at t = 34 hours, the number of cells in the log culture will be P (34) = e 9 ·34 5 ≈ 1150. 2. Make a conjecture for the nth derivative of f (x) = xeax . How might you prove this conjecture? Hint: The first three derivatives are given by: f 0 (x) = eax + xaeax = eax + af (x) f 00 (x) = aeax + af 0 (x) = aeax + a(eax + xaeax ) = 2aeax + a2 f (x) f 000 (x) = 2a2 eax + a2 f 0 (x) = 2a2 eax + a2 (eax + xaeax ) = 3a2 eax + a3 f (x). Can you see a pattern? Note: It is not necessary to write the derivatives in terms of f (x) – I just find that it makes finding the second and third derivatives easier. 3. Differentiate the following functions: (a) f (x) = e−x sin(x). (b) f (x) = ex . ex +e3x (c) f (x) = log(2x) cos(x). (d) f (x) = log(log(log(x))). Hint: The answers are: (a) f 0 (x) = e−x (cos(x) − sin(x)). 2x 2e (b) (Simplify the equation before differentiating) f 0 (x) = − (1+e 2x )2 . (c) f 0 (x) = cos(x) x (d) f 0 (x) = 1 . x log(x) log(log(x)) − log(2x) sin(x). 4. Using properties of the function ex , prove the following identities: (a) log(ab) = log(a) + log(b) for a, b > 0. (b) log(ax ) = x log(a) for a > 0. Solution to (a): We want to use the property that ex+y = ex ey , and that ex and log(x) are inverse functions, i.e. elog(x) = x for all x in the domain of the logarithm function and log(ex ) = x for all x. Note that this tells us that a = elog(a) and b = elog(b) for a, b > 0. So, log(ab) = log elog(a) elog(b) = log elog(a)+log(b) (using the first property) = log(a) + log(b). (using the fact that they’re inverse functions) Proving identity (b) is similar. 2