. ( points) Evaluate each of the following limits. 3x x + (a) lim x x 3 8 x + sin(5x) (b) lim x sin(x) (c) lim x π/3 + sec x ( (d) x x + 5x ) (e) lim x 5 x lim x 5 + x 6. (3 points) What value of c makes the following function continuous at? c x + 3c if x <, f(x) = x if x =, cx + c + if x >. 3. ( points) Give an equation for each horizontal and vertical asymptote of the graph of f(x) = x + sin x 3x +. 4. (4 points) Use the limit definition of the derivative to find f (x), where f(x) = 3x +. 5. ( points) Find dy (a) y = 8 x 3 x + x x (b) y = x + (c) y = cos 3 (6x ) (d) ln(x y) = xy for each of the following. Do not simplify your answer. 6. (4 points) Let f(x) = x+3 9 x (x + ) 4 (6x + 3). Use logarithmic differentiation to find f (). Simplify your answer. 7. (3 points) Use the Intermediate Value Theorem to show that the equation x 3 4x + = has at least one positive solution. 8. (3 points) Find all points P on the parabola given by the equation y = x x such that the line joining P and the point (4, 4) is tangent to the parabola. Page of 5
9. (4 points) (a) State the Mean Value Theorem. (b) Show that if f() = and f (x) 5 for x, then f(4) 8.. (4 points) Consider the curve C given by the equation x 3 + y 3 = 8(xy + ). Find an equation for the tangent line to the curve C at the point (, ).. (6 points) A lighthouse is located on a small island km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam moving along the shoreline when it is.5 km away from P?. (6 points) A right circular cylinder is inscribed in a sphere with radius 3. Find the largest possible volume of such a cylinder. 3 3. (4 points) The position of a particle moving along a straight line at time t is given by s = (t 3) e t where s is measured in meters and t is in seconds. (a) When is the particle at rest? (b) When is the particle moving in the positive direction? 4. (4 points) Find the absolute extrema of f(x) = x + 8 x + 36 on [, 8]. 5. ( points) Given f(x) = 8(x + 4) (x + ), f (x) = (a) x and y intercepts. (b) Vertical and horizontal asymptotes. 3(x ) (x + ) 3 and f (x) = (c) Intervals of which f(x) is increasing or decreasing. (d) Local (relative) extrema. (e) Intervals of upward and downward concavity. (f) Inflection points. ( 64)(x 4) (x + ) 4, find all: (g) Find the coordinates of the point(s) where the graph of f intersects its horizontal asymptote. (h) On the next page, sketch the graph of f(x). Label all intercepts, asymptotes, extrema, and points of inflection. Page of 5
6. ( points) Evaluate each of the following integrals. (a) (e 4x + 3 ) x 5 (b) (c) (d) (x 5 + ) x 4 sec x(sec x + tan x) π/ sin(x) 7. ( points) Find the derivative with respect to x of y = x t + t dt. 8. (5 points) Evaluate (x + ) using the definition of the integral as a limit of Riemann sums. n(n + ) You might use the formulas i =, i n(n + )(n + ) ( ) n(n + ) =, and i 3 =. 6 i= i= i= 9. ( points) Let f be an even function such that (x )f(x). 3 f(x) = 6 and 3 f(x) = 4. Evaluate Answers:. (a) (b) 3 (c) (d) 5 (e). c = 3. One horizontal asymptote; y = /3, and one vertical asymptote; x = /3. 4. f (x) = lim h f(x + h) f(x) h 5. (a) dy = 8 x 3 3 + (ln )x x (b) dy = x (x ) / (x + ) 3/ 3 = = lim h (3x + 3h + )(3x + ) = 3 (3x + ) Page 3 of 5
(c) dy = 36x cos (6x ) sin(6x ) (d) dy = xy + y x xy + 6. f () = 8(ln 6) 7. Let f(x) = x 3 4x +. Note that f() = and f() =. Since f is continuous on [, ] and is between f() and f(), it follows by the Intermediate Value Theorem that there exists a number c in (, ) such that f(c) =. The number c is a positive solution to the given equation. 8. (, ) and (6, 4) 9. (a) If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that f f(b) f(a) (c) =. b a (b) The statement of the problem implies that f is differentiable at all x, and hence f is continuous at all x. Therefore, we may apply the Mean Value Theorem with f and the interval [, 4] to obtain a number c in (, 4) such that f f(4) ( ) (c) =, or equivalently, f(4) = f (c). By the hypothesis f (c) 5, hence f(4) (5) = 8.. y = 5x + 6. 7π km/min. 3π cubic units 3. (a) t = 3 and t = 5 (b) 3 < t < 5 4. The minimum value is 3/5 and the maximum value is. 5. (a) No x-intercept, y-intercepts: (, 8) (b) Vertical asymptote: x =, horizontal asymptote: y = 8 (c) Increasing on (, ) and (, ). Decreasing on (, ). (d) Local minimum: (, 4). No local maximum. (e) Concave down on (4, ). Concave up on (, ), (, 4). (f) Inflection point: (4, 4/9) (g) (, 8) Page 4 of 5
(h) y (, 8) y = 8 (, 4) (4, 4/9) x x = 6. (a) e x 4 ln x + 3 8 x8/3 + C 7. (b) x7 7 + x 3x + C 3 (c) tan x + sec x + C (d) 3 π x 3 + x 4. 8. lim n 9. 96 [ ( ) i + ] n n = 3 i= Page 5 of 5