Math 1071 Final Review Sheet The following are some review questions to help you study. They do not

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1 Math 1071 Final Review Sheet The following are some review questions to help you study. They do not They do The exam represent the entirety of what you could be expected to know on the exam; reflect distribution of questions that will be on the exam; and correspond exactly to the difficulty level of question on the exam. correspond to many types of questions that could be asked; probably contain a few typos or mistakes; and reflect specific requests by students for types of problems they need to practice. will be cumulative; will be slightly weighted toward material covered at the end of the course; may ask about anything covered in class or on homework, unless specifically stated otherwise by your instructor; will be about 16 questions; will look similar to your midterms; and will be on Friday, May 4, at 3:30pm. You can find the room for your section listed in the UConn finals schedule. To study for the exam, go over your class notes, homework exercises, old quizzes, old midterm review sheets, and this review sheet. Ask questions, go to the review sessions, go to class. Study in groups, go to the Q Center, find helpful videos online. Good luck! (1) Find the derivative of the following functions. (a) h(x) = 14 x 12x 7 + x 10 (b) r(x) = x(ln x) (c) u(x) = e 5x (x x e) (d) f(x) = (ln(x 3 + 2x 1))(x 3) 2 (e) g(x) = 3 (x4 7x) + x 2 1 (f) k(x) = e3x2 +5 x+5 (g) q(x) = 13x 2 5x + 8 (h) j(x) = ( ) 8x x x 3 (i) m(x) = 3 log 5 (17 x) + 9 (j) n(x) = 5e x2 7x+12 (k) b(x) = (ln(8x + 1)) 4 (2) Find the slope of the tangent line to f(x) = 5x 2 3 at x = 2. Find the equation of the tangent line at x = 2. (3) Find the tangent line to f(x) = x2 x 3 at x = 4. 1

2 (4) Write the limit definition of the derivative. Find the derivative of f(x) = 2 definition. 3x 1 using the limit (5) Let f(x) = 3x 2 4. Use the limit definition of the derivative to find the instantaneous rate of change at x = 5. (6) What is log 4 16? What is log 4 2? (7) Simplify 10 log 10 3 and log 10 (10 7 ) (notice that log 10 (10 7 ) is not the same as (log 10 10) 7 ). (8) Write 2 log x + log y log z as a single logarithm. (9) Solve the following equation for x: (10) Solve the following equation for x: 2 log 3 (x + 7) + 5 = log 4 16 (11) Solve for x: 5 3x = 125 4x 4 (12) Solve for x: ln(3 4x) ln(1 x) = ln x = 4 (13) Find the absolute extrema of h(x) = x 3 6x 2 on the following intervals: (a) [ 1, 2] (b) [2, 3] (c) [ 1, 5] (d) (, 10] (14) Find the absolute maximum and minimum of h(x) = x 3 x on the following intervals: (a) [ 1, 1] (b) [ 1, 0] (c) [ 1, ) (d) [ 2, ) (15) Find the following limits: (a) lim x x 12 10x 3 4x 7 (b) lim x 6x 9 3x 4 +2x 10 5x 5 4x 9 (c) lim x 3 7 x 2 9 (d) lim x x 6 2x+1 3x+7x 113 (e) lim x 4 + (f) lim x 1 x 7 x 2 8x+16 x+2 x 3 x 2 +1 (g) lim h 0 (2+h) 2 4 h

3 { x if x > 0 (16) Let f(x) = 1 x if x 0. Find lim x 0 f(x), lim x 0 + f(x), and lim x 0 f(x). { x 2 9 (17) Find the following limits if f(x) = 3x 9 if x 3 x 1 if x > 3 (a) lim x 0 + f(x) (b) lim x 0 f(x) (c) lim x 0 f(x) (d) lim x 3 f(x) (e) lim x 3 f(x) (18) If f(x) is given by the picture below, find the following limits: (a) lim x 0 + f(x) (b) lim x 0 f(x) (c) lim x 0 f(x) (d) lim x 2 + f(x) (e) lim x 2 f(x) (f) lim x 2 f(x) (g) lim x 3 + f(x) (h) lim x 3 f(x) (i) lim x 3 f(x) (j) lim x 4 + f(x) (k) lim x 4 f(x) (l) lim x 4 f(x) (m) lim x 5 f(x) (n) lim x f(x) (o) Also, find all values a where lim x a does not exist. (p) Find all values a where f (a) does not exist. (19) Let f(x) = 2x 3 + 3x 2 12x + 6. Sketch a graph of f(x), using what you know from calculus. (20) Let f(x) = x+1 x+2. Sketch a graph of f(x), using what you know from calculus.

4 (21) Let f(x) = x2 +2x 4 x. Sketch a graph of f(x), using what you know from calculus. 2 (22) Say f (x) is given by the graph below. Where is f(x) increasing? Where is f(x) decreasing? Draw a (rough) sketch of f(x). (23) Say f(x) is given by the graph below. Sketch a rough graph of f (x), noting any places where f (x) does not exist. (24) Say the equation f (x) = 0 has two solutions, x = 2 and x = 4. Furthermore, f ( 10) = 7, f (0) = 3, and f (10) = 13. Find any relative extrema of f(x). (25) Find dy dx for the following curves: (a) xy + x 2 = 3y 3 15 (b) x 3 y 3 = 6x + 5 (c) x 2 y y 3 + 3x = 5 (d) x 2 y 2 + xy 4 = 2x 5 (26) Calculate the following indefinite integrals: (a) 2x 3 5 x + 10x e dx (b) x(x + 5) dx (c) x 2 + x 1 + π + x + x 2 dx (d) (2x 3 x)(x 4 x 2 + 6) 5 dx

5 (e) (8x 3 2)(x 4 x + 10) 11 dx (f) 6x 2 2 x 3 x dx (g) ln x dx (h) 3x 2 4x 3x 3 6x 2 dx (i) xe 5x dx (j) xe (x2) dx (k) 8xe 2x2 4π dx (l) (x 1)e x2 2x dx (m) x 5 ln x dx (n) x x 3 dx (o) x 3 dx (p) 1 x ln x dx (27) Calculate the following definite integrals: (a) 1 2x + 1 dx 0 (b) 4 1 x2 x + 1 dx (c) 2 1 (2x 1)(x2 x) 5 dx (d) 3 2 4x x 2 2 dx (e) 1 0 (x 1)ex2 2x dx (f) x dx (28) Find the area enclosed by the given functions: (a) f(x) = x 2 and g(x) = 3x (b) f(x) = x 2 and g(x) = x (c) f(x) = x 3 and g(x) = x 2 (d) f(x) = e x, g(x) = e 2x and h(x) = 7 (e) y = 1 + ln x, y = x 2, x = 1 and x = 4 (f) f(x) = x 3 3x and g(x) = 2x 2 (29) You want to paint a picture. You have enough wood to make a frame that is 400 cm around. What s the largest picture you can make?

6 (30) A fence is to be built around a 300 square foot rectangular field. One side costs twice as much per unit length as the other three. Find the dimensions of the enclosure that minimizes total cost. (31) The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 296 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 4 dollar increase in rent. Similarly, one additional unit will be occupied for each 4 dollar decrease in rent. What rent should the manager charge to maximize revenue? (32) A deli sells 320 sandwiches per day at a price of 4 each. A market survey shows that for every 0.10 reduction in the price, 20 more sandwiches will be sold. How much should the deli charge in order to maximize the revenue? (33) A straight piece of wire 8 feet long is bent into the shape of an L. What is the shortest possible distance between the ends? (34) You are opening a kennel, and creating an area with room for 5 dogs, but the dogs must be kept separate from each other. The outside fencing costs $10 per foot, and inside barriers to separate dogs costs half as much. You have $2000 to spend what s the largest area you can enclose? (Let s suppose all the dogs have an equal amount of space, and you separate them by dividing the area using parallel barriers, something like the following beautiful picture.)

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