Math 121 Sample Problems for Test 3 ex − cos x − 2x = x→0 x2 − 2x 1. lim (A) 1 (B) 0 (C) ∞ (D) 1 2 (F) − (E) 2 1 2 2. An automobile traveling on a straight road applies its brakes and experiences a constant deceleration of 40f t/sec2 while skidding. The car skids 180 feet before coming to a stop. What was its initial velocity? Z 1 a+h 0 3. If f is a continuously differentiable function for all real x, then lim f (x) dx is h→0 h a (a) 0 (b) f (0) (c) f (a) (d) f 0 (0) (e) f 0 (a) Z √x 4. Let k(x) = cos(t2 + 1) dt. Find k 0 (x). 5 5. Compute the following limits: Show all your work. 3 (a) limt→∞ t2 e−t (b) lims→∞ s1/s (c) limθ→0+ 1+sin θ cos θ (a) limn→∞ Pn i=1 (cos xi ) ∆x, 6. Find F 0 (4), where F (x) = 7. If f is continuous and R 24 3 R2 √ x where 0 ≤ x0 < x1 < · · · < xn−1 < xn ≤ π/4 and ∆x = et−2 t dt. f (x) dx = 8, find R2 1 x2 f 3x3 dx. 8. The graph of a function g appears below. Use the Midpoint Rule with n = 3 to approximate 9. Rx R6 0 g(x) dx 2 et dt lim 2 = x→1 x − 1 1 (A) 0 (B) 1 (C) e 2 (D) e (E) DNE 1 π 4n . 10. If g is a differentiable function such that g(x) < 0 for all real numbers x and if f 0 (x) = (x2 − 4)g(x) which of the following is true? 1. f has a relative maximum at x = −2 and a relative minimum at x = 2. 2. f has a relative minimum at x = −2 and a relative maximum at x = 2. 3. f has relative minima at x = −2 and x = 2. 4. f has relative maxima at x = −2 and x = 2. 5. It cannot be determined if f has any relative extrema. 11. Use Newton’s method to find the x coordinate for the inflection point for y = x3 − cos x. (A) xn+1 = x2 = x3 = x4 = x5 = 12. If lim f (x) = 0 and lim f 0 (x) = π, then lim sin πx = f (x) 1 π (E) π 2 x→0 x→0 (A) 1 (B) x→0 (C) π (D) 0 13. Find the derivatives of the functions (i) F (x) = R √x 1 (F) 1 π2 tan−1 (t) dt and (ii) G (x) = R ln(x) √ sin(x) 1 + t2 dt. 14. Let a < c < b and let g be differentiable on [a,b]. Which of the following is NOT necessarily true? (a) Rb a g(x) dx = Rc a g(x) dx + Rb c g(x) dx (b)There exists a d in [a, b] such that g 0 (d) = Rb (c) a g(x) dx ≥ 0 g(b)−g(a) , b−a (d) limx→c g(x) = g(c) (e) If k is a constant, then Rb a k g(x) dx = k Rb a g(x) dx. 15. A publisher estimates that a book will sell at the rate of r(t) = 16, 000e−0.8t books per year at the time t years from now. Find the total number of books that will ever be sold (up to t = ∞). 16. Given a function f with a continuous derivative f 0 on the real line R and with f (2) = 0, f 0 (2) = 5, limx→∞ f (x) = ∞, and limx→∞ f 0 (x) = 3, evaluate the following limits: (a) limx→2 f (x) x2 −4 (b) limx→2 f (x) cos(x) (c) limx→∞ ln(x) f (x) . 2 17. Evaluate the following integrals: R (a) esin x cos x dx (d) R (g) R π/2 x sin (x) dx 0 √ cos x sin x dx R (j) (xex + e1+x ) dx x+1 x2 +2x+5 (b) R (e) R2 (h) R6 √ 4 x x − 4 dx (k) R 0 dx 6x(x2 + 2)2 dx dt 9t2 +4 18. If the function g has a continuous derivative on [0, c], then (a) g(c) − g(0) Rc 0 R (f) R (i) R3 (l) R dx √ x2 1 + x3 dx, 2 (x + x1 )2 dx, x(ln x) dx. g 0 (x) dx = (c) |g(x) − g(0)| (b) g(x) + c 1 (1−4x)2 (c) (d) g(c) (e) none of the above. 19. For a function f with continuous derivative f 0 on the real line, the integral R (a) x3 f x3 − 3x2 f x3 dx R (c) uf 0 (u) du with u = x3 R x3 f 0 x3 dx = R (b) xf x3 − f x3 dx R (d) 31 xf x3 − 13 f x3 dx (e) none of the above. 20. Compute the following or state the integral is divergent. R1 R6 y R2 1 (a) −∞ √3−x (c) 2 √y−2 dx (b) −2 x16 dx dy 21. Compute 2x3 +1 0 x4 +2x+1 dx (a) R 2x−1 dx x2 +1 (b) R1 (d) R x4 ln (x) dx (e) R2 (g) R (h) R x−2 x2 −3x−4 dx 22. Use the Trapezoidal Rule to approximate this approximation? 1 xe−x+1 dx 1 x2 −6x+8 R3√ 0 23. Use Simpson’s Rule with n = 8 to estimate R (f) R sin2 (x) cos3 (x) dx (i) R 1 x2 +6x+25 dx x2 + 1 dx with n = 6. What is the maximum error for Rπ 0 dx √ 3 dx 1−x2 (c) sin2 θ dθ. 24. Consider a tank of water whose volume has a rate of change given by V 0 (t) = 0.8t − 40 (in gallons per minute). What is the net change in the volume of water in the tank after 60 minutes have passed? What is the total volume of water that has been drained or added to the tank after 60 minutes have passed? 25. Use the Riemann sum with left endpoints and n = 6 to estimate the area under the curve y = x sin x on the interval [0, π/2]. Is this an overestimate or an underestimate of the area? Write the value of the area as a limit of a Riemann sum. 3 26. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 3 cm if one side of the rectangle lies on the base of the triangle. 27. A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach form the ground over the fence to the wall of the building? 28. A mathematician wishes to mail popcorn in a cylindrical package of length h and a circular base of radius r. Because of the post office regulations, the length of the cylinder plus the circumference of the base cannot be more than 108 inches. Find the dimensions of the cylindrical package with maximum volume. Justify your answer. 29. A cylindrical can is to be made to hold 1000 cm3 of liquid. Find the dimensions that will minimize the surface area of the can. 30. Suppose that a rectangular box with open top and square base is to be made using two different materials. The material for the base cost $2 per square foot and the material for the four sides costs $1 per square foot. Find the dimensions of the box of greatest volume subject to the condition that $96 is spent for the material. What is the maximum volume? Justify your answer. 31. Amy is setting up a lemonade stand. The cost for making x glasses of lemonade is 5 + 0.02x dollars. Previous experience indicates that she can sell 80 glasses of lemonade at a price of $0.50 per glass and that for each $0.10 increase in price, she will sell 4 fewer glasses. At what price should the lemonade be sold to maximize the profit? 4