In the following exercises, find the power series for the given function using term—by—term differentiation or
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- The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is given by A(t)=a(e)rt, where a is the amount ofprincipal initially deposited into an account thatcompounds continuously. Prove that the percentageof interest earned to principal at any time t can becalculated with the formula I(t)=ert1.arrow_forwardAt time t, in seconds, your velocity, V, In meters per second is given by v(t) = 6 + t², for 0arrow_forwardUse Newton Raphson method to find the root of the function f(x)=D4 In(x)-x with initial guess X, = 8 and perform the computation until | Ea <0.1% .arrow_forwardThe energy E (in J) dissipated by a certain resistor after t seconds is given by E= In (t + 6) - 0.125t. At what time is the energy dissipated the greatest? t= second(s) ...arrow_forwardUse differentials and the graph of f to approximate (a) f (1.9) and (b) f (2.04).arrow_forwardThe inventory for an item in a warehouse is given by I(t) = 90e0.03t +, where t is measured in days since the beginning of 2021. Find the average inventory that was done in the first three months of 2021. t+1arrow_forwardUse differentials to approximate the value of (1.02)².arrow_forwardThe number of people in a community who became infected during an epidemic t weeks after its outbreak is given by the function f(t)= 10,000/(1+ae-kt) where 10,000 people of the community are succeptible to the disease. Assume that 5000 people were infected initially and 8984 had been infected by the end of the fourth week. How many people were infected after 8 weeks?arrow_forwardLet f(x) = /x. Does f(5+ h) equal 5+ h or /5 + /h? Compute the difference quotient at a = 5 with h = 1.arrow_forwardConsider the function f(x) defined by the following rule: a. Suppose you wanted to calculate the derivative, f′(x). What do you get? b. How about f(2), or f (1/3)? Which inputs seem to give a convergent series? c. What if you try to calculate f(1). Does the resulting series converge? What about f (−1)?arrow_forwardSince YouTube first became available to the public in mid-2005, the rate at which video has been uploaded to the site can be approximated by vle) = 1.1r - 2.6 + 2.3 million hours of video per year (0 sts 9), where t is time in years since the start of June 2005.t (a) Find an expression for the total number of hours V(t) of video at time t (starting from zero hours of video at t = 0). (Round your coefficients to four decimal places.) V(e) = | %3D (b) Use the answer to part (a) to estimate the total number of hours of video uploaded by the start of 2007. (Round your answer to the nearest million hours of video.) 20 |× million hours of videoarrow_forwardThe number of new members joining MySpace each year in the period from 2004 to the middle of 2007 can be modeled by m(t) = 10.5t2 + 14t − 6 million members per year (0 ≤ t ≤ 3.5), where t is time in years since the start of 2004.† What was the average number of new members joining MySpace each year from the start of 2004 to the start of 2007? million members per yeararrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios