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- A process creates a radioactive substance at the rate of 1 g/hr, and the substance decays at an hourly rate equal to 1/10 of the mass present (expressed in grams). Assuming that there are initially 20 g, find the mass S(t) of the substance present at time t, and find lim S(t) as t approaches infinityUsing the definition of the derivative, prove that (d/dx)(e^x)= e^x. You will need to use the following limit: lim as t approaches 0 of (e^t-1)/(t)= 1. (In this problem e is just a constant, which happens to be a special number equal to approximately 2.71828...)Let G(t) = (1 - cos t)/t2. a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate lim t-->0 G(t). b. Support your conclusion in part (a) by graphing G near t0 = 0
- 1. Evaluate: lim x→2 (x^2 − 4x + 4)/(tan^2(x2 − 4)) 2. Use the Intermediate Value Theorem to show that f(x) = cos^−1 (x) − e^x has a zero in the interval [0, 1]. 3. Use the Squeeze Theorem to evaluate: lim x→0+ sin(ln x) csch(cot x).1. Find the dy/dx using logarithmic differentiation y=(x2(x4+5)1/2)/(ex-x^4) 2. Find the limit lim x->∞ ((8x3+x2)1/3-2x)Find the Taylor polynomial of degree 3 for sin(x), for x near 0:P3(x)= Approximate sin(x) with P3(x) to simplify the ratio:sin(x)/x= Using this, conclude the limit:lim x→0 sin(x)/x=
- lim as x approaches 0 x sin(x)/1-cos(x)f(x) = { cos x, x< 0 sin x+ 1, 0<- x<-pi cos x, x>pi which lim does not existAssume that V (t) = V is constant and I (0) = 0. (a) Solve for I (t). (b) Show that lim t→∞ I (t) = V/R and that I (t) reaches approximately 63% of its limiting value after L/R seconds. (c) How long does it take for I (t) to reach 90% of its limiting value if R = 500 ohms, L = 4 henries, and V = 20 volts?
- If f(x) = x2+1 for x > or equal to 0, then is arcf(x) has a tangent at x=1For the function f(x) = x 4 ln x (a) Show a careful calculation explaining why limx→0+ f(x) = 0 ( asserting that (0)(−∞) = 0 is not careful enough) (b) Calculate a formula for f 0 (x), the derivative of f. (c) Find the critical number(s) for the function f. (d) Determine the absolute maximum and absolute minimum values of f on the interval 1 2 ≤ x ≤ 1, and the exact x values at which these maximum & minimum values occur. Please solve part b c D with a detailed solution.Find the derivative g(t)=1/(t^1/2) using the limit definition of a derivative. Check your answer by computing the derivative using tools we have now. Compute the derivative of h(x)=x^1/3 (equivalent to cubed root of x). Show that h'(0) does not exist, and determine what the tangent line to the cubed root of x looks like at x=0