Please help with following question:

1. Evaluate the integral. dt (a) 4t + 3 (b) / 3x267+1 dx 1 (c) dx Note: this is an improper integral. x VIn x (d) dx V1 - e2x 8x - 1 (e) dx 2x2 - 3x - 2 (f ) x2 + 3x +1 dx x2 +4 2. Find the limit. (a lim x-0 e3x - 3x - 1 (b) lim (1 - 5x) 23. Consider the function f (9:) : 17629:. (a) On what open intervals is f(3:) increasing/ decreasing? Find all local extrema of f (SC) Justify your answer. (b) On what interval or intervals is f (as) concave up/concave down? Find all inflection points of x). Justify your answer. (0) Find Ilirgrjx). ((1) Find Em f($). 4. Consider the differential equation Let y : f (1:) be the particular solution to the differential equation with initial condition f (0) : 1. (a) Find lim &. m>0 23: 1 (b) Use the tangent line to the graph of y 2 x) at m = 0 to estimate f(0.5). (C) Is the graph of f is concave up or concave down at (0, 1)? Is the approximation in part b) an overestimate or underestimate? ((1) Use Euler's method, starting with ct: = 0 with two steps of equal size, to approximate f (1) (e) Find 3; = f (m) the particular solution to the differential equation with initial condition f(0) = 1 5. A population is modeled by the function P that satises the logistic differential equation dP P P - *(1 ) 35 _E (a) If P(0) = 3, what is tlirn P05)? 00 If P(0) = 20, what is flirn P(t)? )30 (b) If P (0) = 3, for what value of P is the population growing the fastest? If P (0) = 7, for what value of P is the population growing the fastest? (o) A different population is modeled by a function Y that satises the separable differential equation : Z (1 _ i) alt 5 12 Find Y(t) if Y(O) = 3. (d) For the function Y found in part (Q), what is lirn Y(t)? tAoo 6. Match the slope fields with their differential equations. I. y' = 1+yz II. y = y' III. y' = cos x IV y' = 4 - y