Download Math 201-103-RE: Winter 2011 Final Exam - Limits, Derivatives, and Applications and more Exams Calculus in PDF only on Docsity!
(Marks) (4) 1. Use the graph to find the following limits. Use โ, โโ, or DNE where appropriate.
(a) (^) xlimโโ 2 f (x) =
(b) (^) xโโโlim f (x) =
(c) (^) xlimโโ f (x) =
(d) lim xโ 0 โ^ f (x) =
(e) lim xโ 2 f (x) =
(f) lim xโ 0 +^ f (x) =
(g) f (0) = (h) f (2) =
x
y
โ 2 0 2 โ 2
3
1
(15) 2. Use algebraic techniques to evaluate the following limits. Identify the limits that do not exist, and use โ or โโ where appropriate. Show your work.
(a) (^) xlimโโ 2 2 x^2 + x โ 6 3 x^2 โ 12
(b) lim xโ 7
x โ 3 x โ 7
(c) (^) xโโโlim 4 x^4 โ 3 x^3 โ 4 โ 2 x^5 + 1 โ 5 x
(d) lim xโ 0 โ
x
x^2
(e) lim xโ 3 +^ f (x) where f (x) =
x^2 + 3 for x < 3
x โ 3 x^2 โ 9 for x > 3
(3) 3. Use the definition of continuity to find the value(s) of x for which the following function is discontin- uous.
f (x) =
x^2 โ 3 for x < โ 2
1 (2x + 7)(x โ 4)
for x โฅ โ 2
(3) 4. Find the value(s) of k such that f (x) is continuous for all real numbers
(Marks)
f (x) =
โx^2 โ 5 k for x < 2
k^2 โ
x
for x โฅ 2
(5) 5. (a) Use the limit definition of the derivative to find f โฒ(x) if f (x) =
3 x โ 8. (b) Check your answer using the derivative rules. (c) Use your answer in part a) to find the slope of the line tangent to f (x) at x = 4.
(28) 6. Find dy dx for each of the following. Do not simplify your answers.
(a) y =
x
x + x^4 โ 4 x
(b) y = 5x log 2 (sin x)
(c) y =
e^3 x+ sec 3x
(d) y = ln
(3x โ 2)^5 (4 โ 2 x)^6
(e) y = x^3 cos^2 x + x^3 sin^2 x + ฯ (f) y = 5 (3x)e x
(g) 4x^2 y^3 + x^3 = (3x + y)^2
(4) 7. Determine the xโvalue(s) where f (x) has horizontal tangents given f (x) =
4 x^2 + 7x โ 2
(4) 8. Given the function f (x) = e^3 x^ cos (1 + x) , determine f โฒโฒ(0).
(4) 9. Use the second derivative test to determine the relative extrema of f (x) = 3x^3 โ 9 x (4) 10. Determine the absolute maximum and minimum of f (x) = x^2 eโx^ on the interval [โ 1 , 1].
(10) 11. Given f (x) = 3 x^2 x โ 1 ; f โฒ(x) = 3 x(x โ 2) (x โ 1)^2 ; f โฒโฒ(x) =
(x โ 1)^3
(a) Find the yโintercept, xโintercept, any vertical and horizontal asymptotes, relative extrema and points of inflection (if any). Find the intervals where f is increasing, decreasing, concave up and concave down. (b) Sketch a graph of f (x).
(Marks) Answers
- a) 0 b) +โ c) 1 d) โ 2 e) +โ f) 0 g) 3 h) undefined
- a) 127 b) โ^14 c) 0 d) โโ e) (^16)
- x = โ2 or x = 4 4. k = โ6 ; k = 1
- a) Use f โฒ(x) = lim hโ 0
f (x + h) โ f (x) h b)f โฒ(x) =
3 x โ 8
c)f โฒ(4) = (^34)
- a) dy dx = โ 2 xโ^3 /^2 โ 14 xโ^3 /^4 + 4x^3 โ 4 x^ ln(4) b) dy dx = 5 log 2 (sin x) + 5x. cos x sin x ln(2)
c) dy dx
e^3 x+ sec 3x
)โ 1 / 2 [
3 e^3 x+2^ sec 3xโ3 sec 3x tan 3x e^3 x+ sec^2 3 x
]
d) dy dx
3 x โ 2
4 โ 2 x
e) dy dx
= 3x^2 cos^2 x โ 2 sin x cos x. x^3 + 3x^2 sin^2 x + 2 sin x cos x. x^3 f) dy dx
= 5 (3x)e x
[
ex^ ln(3x) + ex x
]
g) dy dx
18 x + 6y โ 8 xy^3 โ 3 x^2 12 x^2 y^2 โ 6 x โ 2 y
- x = โ^78 8. 8 cos(1) โ 6 sin(1) โ โ 0. 73
- relative maximum at (โ 1 , 6) and relative minimum at (1, โ6)
- absolute maximum is 2.72 at x = โ1 and absolute minimum is 0 at x = 0
- a) y-int and x-int:(0,0) vertical asymptote: x = 1 horizontal asymptote: none relative maximum: (0, 0) relative minimum: (2, 12) PI: none Inc: (โโ, 0) โช (2, +โ) Dec: (0, 1) โช (1, 2) CU: (1, +โ) CD: (โโ, 1)
- a) 105 people
- b) maximum revenue is $551. 25
- cost of materials: $
- a) ฮท(300) = โ 1. 46 b) elastic at x = 300 units; if price decreases by 10%, the quantity increases by 14.6%; revenue will increase c) yes, unit elasticity at x = 400 units
11 b) f^ (x)
(^01 2) x
12
