Find each limit. Show all steps and algebra in completing each problem. (a) lim x→  x+3 2 x2 − 1

Math 1A Final Exam Name_________________________ Show all work for full credit. 1. Find each limit. Show all steps and algebra in completing each problem. (a) lim x→  x+3 2 x2 − 1 (b) lim 4 x 2 + 3x − 2 x x→  (c) lim x→1 x sin( x − 1) 2×2 − x − 1 (d) lim+ (1 − cos x)sin x x→ 0 2. Find the derivative of each. Show all steps. You do NOT have to simplify your answers. (a) y = e 1 x (b) y = log x 2 − cos x 5sec x 2 x2 − 1 (c) cot( x − y ) = 2 xy 3 − 1 (d) y = ( sin x ) ln x 3. Find the most general antiderivative of the following: (a) f ( x) = −3 x 5 + 4 x 7 − 14e x − ln 2 ( 2 x ) − sec x tan x − 1 + 2 3×5 1 2 4. Find the critical numbers of the function f ( x) = x 3 (4 − x) 3 . 5. Find the value of the differential dy if y = answer to two decimal places. x−5 if x = 1 and dx = 0.3 . Round your 2x + 3 6. The angle of elevation is the angle formed by a horizontal line and a line joining the observer’s eye to an object above the horizontal line. A person is 100 feet way from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 15 ft/sec. At what rate is the angle of elevation, θ, changing when the hot air balloon is 50 feet above the ground. Round answer to 3 decimal places. 100 ft 7. Find the root to the function f ( x) = arctan x − x − 1 using Newton’s Method correct to 6 decimal places. Use x1 = 2 . Show all work and show the result at each step. 8. A rectangular storage container with no top is to have a volume of 10 m3. The length of the base is twice the width. Material or the base costs $10 per square meter while materials for the sides costs $6 per square meter. Find the absolute minimal cost to build such a container. Be sure to find the domain and prove your answer. 9. Suppose a car is traveling at a rate of 60 ft/sec on the highway. The driver suddenly sees something and applies the brakes, causing a constant deceleration of -20 ft/sec2. How far will the car travel after the driver hits the brakes until the car stops completely? Round your answer to two decimal places. 10. Eliminate the parameter and graph the parametric curve given by the following set o parametric equations. Be sure to indicate the direction of the path. x = e−t y = et 11. (BONUS) (a) Find the equation of the tangent line to the curve given by the set of parametric equations x = t3 + 1 at the point where t = −1 . d2y (b) Find at the same point, where t = −1 . dx 2 y = t4 + t

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