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(Marks) (15) 1. Use algebraic techniques to evaluate the following limits. Identify the limits that do not exist and use โโ or โ as appropriate. Show your work.
(a) lim xโ 3
x^2 โ 5 x + 6 x^2 โ 9
(b) lim xโ 5
x โ 5 โ x + 4 โ 3
(c) lim xโโ 2 +
x โ 2 x + 2
(d) lim xโโ 2
4 x+6 โ^1 x + 2
(e) lim xโโโ
4 x^4 + 3x^2 + 2 5 x^3 โ 2 x + 7
(f) lim xโ 4 โ
2 |x โ 4 | x โ 4
(4) 2. Use the graph of the function f (x) below to find the following. Use โ, โโ, or DNE where appropriate.
(a) lim xโโโ f (x) =
(b) lim xโโ 1 f (x) =
(c) lim xโ 2 โ^
f (x) =
(d) lim xโ 2 +^
f (x) =
(e) lim xโ+โ f (x) =
(f) lim xโโ 4 f (x) =
(g) f (โ1) =
(h) f (โ4) =
x
y
โ 4 โ 3 โ 1 0 1 2
1
2
3
(3) 3. Find the point(s) of discontinuity of the function. Justify using the definition of continuity.
f (x) =
x+ (xโ5)(x+2) if^ x <^1
โ 2 x+5 if^ x^ โฅ^1
(3) 4. Find the value(s) of the constant k such that the following function f (x) is continuous for all real numbers.
f (x) =
x^2 + k^2 x if x โค 1
5 k + 7x if x > 1
(Marks) (5) 5. (a) State the limit definition for the derivative of a function f (x).
(b) Use the above definition to find the derivative of f (x) =
2 โ 3 x
(c) Use derivative rules to check your answer from (b).
(5) 6. Given x^2 y^2 = (x + y)^2 โ 5 (a) Find yโฒ (b) Find the equation of the tangent line to the curve at the point (1, 2).
(27) 7. Find the derivative for each of the following functions. Do not simplify your answers.
(a) y = 7x^2 โ 3
x + 2xe^ +
x
(b) y = e^3 โ^4 x^ csc(5x)
(c) y = ln
x^5 ยท(2xโ1)^4 tan^6 (x)
(d) y = x (^2) + x^3 +xโ 1
(e) y =
ex^ + sin(x^2 )
(f) y = 1+cot(2 1 โln(xx)) (g) y = 5x^ cos(x^5 ) (h) y = (x + 1)x
2
(10) 8. Given f (x) =
3 x^2 x^2 + 3
with f โฒ(x) =
18 x (x^2 + 3)^2
and f โฒโฒ(x) =
54(1 โ x^2 ) (x^2 + 3)^3
(a) List, if any, x and y intercepts, vertical and horizontal asymptotes, intervals where f (x) is increasing and decreasing, relative extrema, intervals where f (x) is concave up and concave down, points of inflection. (b) Sketch a labelled graph of f (x).
(4) 9. Use the second derivative test to find all relative (local) extrema of f (x) = x^4 โ 18 x^2 + 5.
(4) 10. Find the absolute (global) extrema of f (x) = x^3 + 3x^2 โ 9 x + 2 on the interval [โ 2 , 2].
(Marks) Answers
- (a)1/6 (b)6 (c)โโ (d)โ 1 / 4 (e)โโ (f)โ 2
- (a)โโ (b)1 (c)3 (d)โโ (e)1 (f)D.N.E. (g)2 (h)
- x = โ2 (be sure to justify) 4. k = โ 1 , 6
- (a)f โฒ(x) = lim hโ 0
f (x + h) โ f (x) h
(b)f โฒ(x) =
(2 โ 3 x)^2
(c) use quotient or chain rules
- (a)yโฒ^ = 2(x+y)โ^2 xy
2 2 x^2 yโ2(x+y) (b)^ y^ =^ x^ + 1
- (a)yโฒ^ = 14x โ 13 xโ^2 /^3 + 2e xeโ^1 โ xโ^3 /^2 (b)yโฒ^ = e^3 โ^4 x(โ4) csc(5x) + e^3 โ^4 x
โ csc(5x) cot(5x)
(c)yโฒ^ = (^) x^5 + (^2) x^8 โ 1 โ 6 sec
(^2) (x) tan(x) (d)y
โฒ (^) = 2 x(x^3 +xโ1)โ(3x^2 +1)(x^2 +1) (x^3 +xโ1)^2
(e)yโฒ^ = 4
ex^ + sin(x^2 )
ex^ + 2x cos(x^2 )
(f)yโฒ^ = โ csc^2 (2x)2(1โln(x))โ(1+cot(2x))(โ (^1) x ) (1โln(x))^2 (g)y
โฒ (^) = 5x (^) ln(5) cos(x (^5) ) + 5x(โ sin(x (^5) )5x 4 )
(h)yโฒ^ = (x + 1)x 2
2 x ln(x + 1) +
x^2 x + 1
- (a) x-int:(0,0) y-int:(0,0) VA: none HA:y = 3 Dec:(โโ, 0) Inc:(0, โ) Rel. Min:(0,0) CU:(โ 1 , 1) CD:(โโ, โ1) โช (1, โ) IP:(1,0.75) ; (โ1,0.75) (b)
x
y
โ 1 1
1
3
- Rel. Max.:(0,5) Rel. Min.:(3, โ76) and (โ 3 , โ76)
- Abs. Min.:(1, โ3) Abs. Max.:(โ 2 , 24)
- $
- dimensions are 750 by 1500 meters
- (a)ฮท = 2(xโ x 81); ฮท(65) = โ 0. 49 (b)since |ฮท(65)| = |โ 0. 49 | < 1 inelastic (c)x = 54 for unit elasticity 14 (a)Cโฒ(x) = x^2 + 120x + 500 (b)R(x) = 23 x^3 + 15x^2 + 2500x (c)P (x) = 13 x^3 โ 45 x^2 + 2000x (d) 40 cases of cookies
