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Calculus Review272415063500Let fx=x2+1, x<2-x+3, x≥2Determine the following limit limx→2f(x)Find the limit of limx→-3(2x2+6x)Suppose that limx→cf(x)=8 and limx→cg(x)=13. Find the following limit.limx→c[fx+gx]Find the following limit (if it exists). Write a simpler function that agrees with the given at all but one point.limx→89x2-57x-120x-8Find the limit (if it exists).limx→2-f(x) where fx=-x2-8, x<2x+2, x≥2Find the limitlimx→∞6+5x3Find all vertical asymptotes (if any) of the graph of the function fx=3(x+2)3Find the limit (if it exists). Note that fx=x represents the greatest integer function.limx→3+5x-[x]276225028067000Use the graph as shown to determine the following limits, and discuss the continuity of the function at x=2.i limx→2+fx ii limx→2-fx iii limx→2f(x)Find the x-values (if any) at which fx=x-3x2-3x is not continuous. Find the value of c guaranteed by the Intermediate Value Theorem.fx=x2-2x-4, 4,8, fc=11Locate the absolute extrema of the function gx=5x+46 on the closed interval [0,4] Find the derivative of the function gx=7 by the limit process.The graph of the function f is given below. Graph f’. Use the product rule to differentiate fx=x-4cosx Use the rules of differentiation to find the derivative of the function gx=x7+4x5Find the derivative of the function ft=1+8t63Find dydx by implicit differentiation given that 5xy=8.Find dydx at the point (2,1) for the equation x=y3-7y2+8. Suppose that an automobile’s velocity starting from rest is vt=200t5t+15 where v is measured in feet per second. Find the acceleration at 5 seconds. Round your answer to one decimal place.Use logarithmic differentiation to find dydx for y=5x-34x+22The height of an object t seconds after it is dropped from a height of 550 meters is st=-4.9t2+550. Find the time during the first 6 seconds of fall at which the instantaneous velocity equals the average velocity.427672513843000Describe the x-values at which the graph of the function f(x)=x3x2-9 shown is differentiable. Find the slope of the graph of the function at the given value.fx=5x2+5x-9x2 when x=4 Find an equation of the tangent line to the graph of f at the given point.fv=v-4v2-6, at (3,-3)For the function fx=2x3-12x2+6(a) Find the critical numbers (if any)(b) Find the open intervals on which the function is increasing or decreasing(c) Identify all relative extrema.Find the points of inflection and discuss the concavity of the function fx=sinx+cosx on the interval (0,2π).404812538417500Find two positive numbers such that the sum of the first and twice the second is 120 and whose product is a maximum.Use the graph of f’ to sketch the graph of f. A ball is thrown straight down from the top of a 260-ft building with an initial velocity of -30 ft per second. The position function is st=-16t2+v0t+s0. What is the velocity of the ball after 2 seconds?A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changed when the base of the ladder is 12 feet from the wall. Round your answer to two decimal places.Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function 3x2-x-1 and the x-axis over the interval [3,15]. Round your answer to the nearest integer. 45053252286000The diagram shows the upper and lower sums for the function fx=4x+1 using 6 subintervals. Use upper and lower sums to approximate the area of the region using 6 subintervals.Write the following limit as a definite integral on the interval [4,6] where ci is any point in the ith subinterval.lim?x→0i=1n-5ci+3?xiEvaluate the definite integral 376t2dt. Use a graphing utility to verify your results.Evaluate the definite integral of the function 06π-2z+2coszdz.Find the indefinite integral x73+x85.Use the trapezoidal rule to estimate the number of square meters of land in a lot where x and y are measured in meters. The land is bounded by a stream and two straight roads that meet at right angles.X050100150200250300350400Y1801401201109075706540A ball is thrown vertically upward from a height of 9 ft with an initial velocity of 70 ft per second. How high will the ball go? Note that the acceleration of the ball is given by at=-32 feet per second per second.Suppose that the population (in millions) of Egypt in 2007 was 80.3 and that the expected continuious annual rate of change of the population is 0.017. Find the exponential grown model P=Cekt for the population by letting t=0 correspond to 2000. Round your answer to three decimal places.43529251905000Sketch a few solutions of the differential equation on the slope field and find the general solution analytically. dydx=-7x3y Set up the definite integral that gives the area of the region bounded by the graph of y1=x2+x+1 and y2=x+5.Find the average value of the function fx=24-6x2 over the interval -5≤x≤5.The base of a solid is bounded by y=3x3, y=0, and x=2. Find the volume of the solid if its cross sections are squares (taken perpendicular to the y-axis).Suppose that R1=5.21+0.66t and R2=5.21+0.38t model the revenue (in billions of dollars) for a large corporation. The model R1 gives projected annual revenues from 2008 to 2013, with t =8 corresponding to 2008, and R2 gives projected revenues if there is a decrease in the rate of growth of corporate sales over the period. Approximate the total reduction in revenue if corporate sales are actually closer to the model R2. Round your answer to three decimal places. ................
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