Math 111 Calculus I Fall 2005 Practice Problems For Final December 5, 2005

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Randolph Spencer
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1 Math 111 Calculus I Fall 2005 Practice Problems For Final December 5, 2005 As always, the standard disclaimers apply In particular, I make no claims that all the material which will be on the exam is represented in these problems, and I make no claims that these problems will take the same amount of time as those on the exam either individually or collectively. But if you can do all of these problems without the help of a buddy or your book, you should be in pretty good shape for the final exam. Good luck in your studies! 1. Find the following limits. If the limit does not exists explain why. You must justify your answers with some algebraic work. (a) lim x 0 + x 2 x (b) lim h 0 (h 1) 2 1 h 2. Let f be defined as follows: (a) Find (b) Find lim x 2 f(x) lim f(x) x 2 + (c) Find lim x 2 f(x) x if x 2 f(x) = x 2 + x + 1 if 2 < x < 1 x + 2 if x 1 (d) Find lim x 1 f(x) (e) Find lim x 1 + f(x) (f) Find lim x 1 f(x) (g) At what values of x is the function discontinuous?

2 3. Given the following information about limits, sketch a graph which could be the graph of y = f(x). Label all horizontal and vertical asymptotes. lim f(x) =, lim f(x) = x 1 x f(0) = 0 is an inflection point f (x) > 0 when x > 1 f (x) does not exist at x = 2 f(x) is increasing when x < 2 f (x) < 0 when 2 < x <. Use the limit defintion to find the derivative of f(x) = x + 1 x Find the derivative ( (a) y = x ) 5 x ( ) tan x (b) y = ln x ln t (c) sin(πt) (d) (cos x) x (e) y x x y = Given f(x) = x, find the equation of the tangent line(s) to the graph of y = f(x) x 1 and parallel to y = 3x A particle moves along a straight line. Its position is given by y = t 2 + 7, where y is measured in feet and t is measured in seconds. Find the acceleration of the particle when t = Show that if f(x) = cot x, then f (x) = 1 + x Using the definition of derviative, calculate the derivative of y = x 2 3x. 10. Use the Intermediate Value Theorem to show that the equation cos x = x has a root. 2

3 11. Suppose f is the function whose graph is below. y y = f(x) x 2 f(x) f(3) (a) Let h(x) = and g(x) = x 3 are positive, negative, or zero. x 3 f(t) dt. Decide if the following quantities i. f (3) ii. f ( 3) iii. h() iv. h(5) v. g() vi. g(0) (b) Find g (1). (c) Find 2 f (x) dx. (Note: you are integrating f not f.) (d) Recall the statement of the Mean Value Theorem. On the graph of f above, illustrate the application of the Mean Value Theorem on the interval [ 2, 2]. State, in words, the fact you are trying to demonstrate on the graph. (e) Find the interval(s) where y = g(x) is concave down. If there are none, be sure to say so. 12. Find the smallest slope of the tangent line to the graph of y = 2x 3 2x 2 + 7x + 5. Calculators are not allowed on this problem. 13. Prove that if a differentiable function f(x) has three roots then there is some point so that f (x) = Find the point on the parabola y = 2x 2 which is closest to the point (2, 5). 3

4 15. Let g(x) = ln(x + 8). Find any asymptotes of g(x). Find the intervals on which g(x) is increasing. Find the intervals on which it is decreasing. Find the intervals where g(x) is concave up. Find the intervals on which g(x) is concave down. Using the above information, sketch a graph of g(x), making sure to label all critical points and points of inflection. 16. Let F (x) = 100 x ex + 1. Find F (x). 17. Let f(x) and g(x) be functions so that 5 0 f(x)dx = 10, 5 2 g(x)dx = 5 and 5 0 [f(x) 2g(x)]dx = 0. What is 2 0 g(x)dx? 18. Find the following integrals. Calculators are not allowed on this problem. (a) 2x x dx (b) 1 0 5x dx (c) 1 2 0(2x + 1 2x )dx 19. Evaluate the definite integrals and find the indefinite one: (a) (b) (c) (d) π (x π + π x + π π ) dx x 3 cos x 2 dx ( ) 16 x 2 dx 7 x dx 20. Use the definition of the derivative as a limit to compute f (x) for f(x) = x The area of a circular puddle is growing at a rate of 12 cm2/s. How fast is the radius growing at the instant when it equals 10 cm?

5 22. Compute the following derivatives using the values given in the table below. f(x) f (x) g(x) g (x) h(x) h (x) x = 1/2 π 3 2 π/2 2/3 x = 1 π 2 /3 e 1/2 7 2e x = 2/3 π/3 /5 1 1/e π/e (a) F (1) where F (x) = ( ln g(x) ) 2, (b) G (1/2) where G(x) = f(x) sin(h(x)), (c) H (2/3) where H(x) = e g(x) + 3 x. 3/2 23. Let f(x) be a function with derivative f (x) = cos(x 2 ) and f(1) = 3. Find the linearization of f(x) at a = 1 and use it to approximate f(1.1). Is this approximation an overestimate or an underestimate? 2. A landscape architect plans to enclose a 3000 square foot rectangular region in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $10 per foot along the fourth side. Find the minimum total cost. 25. Suppose one rock is thrown upward from a cliff 25 meters above the ground on Earth with speed 3 m/s and another rock is thrown upward simutaneously from a cliff 16 meters above the ground on the Moon with speed 5 m/s. Acceleration due to gravity on the moon is 1.6 m/s 2. (a) Which rock reaches the highest height above their respective grounds? (b) Will the two rocks ever be at the exact same height above the ground at the same time? If so, at what time t will this occur and are the rocks moving upward or downward at this time? (c) Which rock hits their respective ground first? 26. Arvin is perched precariously the top of a 10-foot ladder leaning against the back wall of an apartment building (spying on an enemy of his) when it starts to slide down the wall at a rate of ft per minute. Arvin s accomplice, Jack, is standing on the ground 6 ft. away from the wall. How fast is the base of the ladder moving when it hits Jack? 27. Let F (x) = 1 0 (sin( t)dt. What is F (x)? 5

Math 180, Lowman, Summer 2008, Old Exam Problems 1 Limit Problems

Math 180, Lowman, Summer 2008, Old Exam Problems 1 Limit Problems 1. Find the limit of f(x) = (sin x) x x 3 as x 0. 2. Use L Hopital s Rule to calculate lim x 2 x 3 2x 2 x+2 x 2 4. 3. Given the function