Find the values of a and b in the function P(t)=abt that models the number of bacteria in a lab experiment, given the function P(t)=47(1.112)5t. Round the final values of a and b to 4 decimal places.
Soit la fonction g(x)=ex(1−x)−1 définie sur R. Étudier le sens de variation et le signe de g. Soit la fonction f(x)=ex−1x+2 pour x=0, et f(0)=3. Déterminer les limites de f, étudier son sens de variation, et trouver l'équation de la tangente à (E) à l'origine. Soit h(x)=f(x)−x, montrer que h′(x)<0 et en déduire qu'il existe une solution unique α à l'équation f(x)=x dans l'intervalle ]2;25[.
1. Find the antiderivative of f(x):
a) f(x)=4x3+3x4
b) f(x)=101x4−x410
e) f(x)=2ex+1
f) f(x)=31ex−2x2 2. Find the antiderivative of f(x):
a) f(x)=(3x−1)2
b) f(x)=(−5x+1)43
e) f(x)=e4x−1
f) f(x)=52e2−5x
i) f(x)=6x
j) f(x)=9−31x
A snowball's radius decreases at 0.1 cm/min. Find the rate of volume decrease when radius is 11 cm. (Round answer to 3 decimal places) dtdV=−4πr2dtdr=−4π(11)2(0.1)=−12.116mincm3
Find the average velocity of an object moving along a vertical line with position L(t)=−3t2+t+8 (m) at time t (s) for the given intervals: [3s, 9s], [2s, 7s], [5s, 8s].
Find the radius of convergence R of the power series ∑n=1∞n(−4)n(x+8)n. If R is infinite, type "infinity" or "inf". Answer: R=41. What is the interval of convergence? Answer (in interval notation): (−12,−4).
Analyze the properties of the logarithmic functions y=log2(x+1) and y=log(x)−3, including domain, range, asymptotes, x-intercepts, and transformations.
Use Newton's method to find the solution to e−2x=−3x+9 starting with x0=−5. Provide the first two iterates x1 and x2, and the final solution accurate to 4 decimal places.
Find the limit, continuity, and type of discontinuity of the piecewise function f(x) = \\begin{cases} 2x+1, & x>-1 \\\\ x^2+1, & x \\leq-1 \\end{cases} at x=a.