Download Math 201-103-RE: Fall 2009 Final Exam - Limits, Derivatives, and Applications and more Exams Calculus in PDF only on Docsity!
(Marks) (12) 1. Use algebraic techniques to evaluate the following limits. Identify the limits that do not exist and use −∞ or ∞ as appropriate. Show your work.
(a) (^) xlim→− 3 x
(^2) − 2 x − 15 x + 3
(b) (^) xlim→− 2
x + 3 − 1 x + 2 (c) (^) x→−∞lim
2 x − 6 x + 1 (d) (^) xlim→∞
3 + √^5 x
(e) (^) xlim→ 2 − f (x), where f (x) =
3 x − 1 if x < 2 x^2 + 4 if x ≥ 2
(f) (^) x→−lim 3 + x (^2) + 6^ x^ + 2x + 9
(4) 2. Use the graph of the function f (x) below to find the following. Use ∞, −∞, or DNE where appropriate.
(a) (^) x→−∞lim f (x) =
(b) (^) x→−lim 3 − f (x) =
(c) (^) x→−lim 3 + f (x) =
(d) (^) xlim→ 1 − f (x) =
(e) (^) xlim→ 1 + f (x) =
(f) (^) xlim→∞ f (x) =
(g) f (−2) = (h) f (1) =
x
y
− 3
1
− 3
− 1
1
2
(3) 3. For the function f (x) as defined in question number 2, (a) list the value(s) of x where f (x) is discontinuous, (b) list the value(s) of x where f (x) is continuous but not differentiable. (2) 4. Find the value(s) of x for which the following function is not continuous. Justify using the definition of continuity. f (x) =
24 − x^2 if x ≤ 5 | 4 − x| if x > 5
(Marks) (2) 5. Find the value(s) of k that will make f (x) continuous for all real numbers.
f (x) =
x^2 + 2x − 3 x − 1 if^ x^ = 1
k^2 if x = 1
(8) 6. Given f (x) =
5 − x , (a) find f ′(x) using the limit definition of the derivative, (b) find the equation of the tangent line to f (x) at x = 1.
(24) 7. Find
dy dx for each of the following functions.^ Do not simplify your answers. (a) y = 5x
x^2 + 1
(b) y =^6 x
2 x^1 /^2 +^
2 x −^ √e x
(c) y = 3 sin(3− 2 cosx) x (d) y = tan(2x) + esec 3x
(e) y = ln
cos x √ (^3) x (^3) + 2 (3x − 1) 2
(f) y = log 7 (x^4 − 3 x) + 2^3 x (g) ey^ = x^2 y^3 + 4 (h) y = (3x^2 + 5)
√x
(3) 8. Given the function f (x) = x sin(2x) , find f ′′(0). (4) 9. Find the x-values of the points on the graph of f (x) = 23 x^3 − 12 x^2 − x where the slope of the tangent line is 2. (4) 10. Find the absolute extrema of f (x) = 2x^3 − 9 x^2 + 3 on the interval [2, 5].
(10) 11. Given f (x) = 2 x
2 (x − 1)^2 f^
′(x) = −^4 x (x − 1)^3 f^
′′(x) = 4(2x^ + 1) (x − 1)^4
(a) find all x and y intercepts, vertical and horizontal asymptotes, intervals where f (x) is increasing or decreasing, relative extrema, intervals where f (x) is concave up or concave down, and points of inflection. (b) sketch the graph of f (x) on the following page.
(4) 12. Given f (x) = 3x^4 − 18 x^2 + 24x + 5 f ′(x) = 12(x − 1)^2 (x + 2) f ′′(x) = 36x^2 − 36 find all relative extrema of f (x). (5) 13. A car dealer sells 80 cars per month at a price of $20 000 per car. For every decrease of $1000 in the price, 10 more cars are sold. What is the price to maximize the revenue? (Be sure to use a test to confirm that this is a maximum.) (4) 14. The average cost function in $/unit is given by C = − 2 x^2 + 3x + (^10) x where x is the number of units produced. Find the marginal cost at x = 50 and interpret the result.
(Marks) (11)
x and y intercept: (0,0) vertical asymptote: x = 1 horizontal asymptote: y = 2 relative minimum: (0, 0) points of inflection:
−^12 , 29
increasing: 0 < x < 1 decreasing: x < 0 or x > 1 concave up: − 12 < x < 1 or x > 1 concave down: x < −^12
x
y
0 1 − 1
1
2
(12) relative minimum at (− 2 , −67) ; (13) the price is $14,000 to maximize the revenue
(14) the marginal cost is C′(50) = −14 700 dollars/unit ; C′(50) ≈ C(51) − C(50)
(15) the dimensions are 25 by 50 meters to maximize the area.
(16a) η = −x
x^2
; (16b) η(3) = −^16 9
≈ − 1 .78 ; (16c) demand is elastic
