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(Marks) (4) 1. Use the graph of the function below to find the following limits. Use โ, โโ or DNE where appropriate.
(a) (^) xโโโlim f (x) =
(b) lim xโ 2 โ^ f (x) =
(c) lim xโ 6 +^ f (x) =
(d) lim xโ 1 โ^ f (x) =
(e) lim xโ 1 f (x) =
(f) (^) xlimโโ f (x) =
(g) f (1) =
(h) f (2) =
x
y
(^0 2 4 )
1
2
3
(15) 2. 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 2 x^2 โ 4 โx^2 โ 5 x + 14 (b) lim xโ 3 โ^ f (x), where f (x) =
7 โ x^2 if x < 3 2 x โ 4 if x โฅ 3
(c) (^) xโlim+โ (2x โ 1)(x + 2) x โ 5 (d) lim xโ 5 +
4 x โ 2 2 x^2 โ 7 x โ 15
(e) lim xโ 3
x + 1 โ 2 x โ 3
(Marks) (3) 3. Given the graph of y = f (x)
(a) Give the interval(s) where the slope of the tangent line to the curve of f (x) is negative.
(b) Locate the x-value(s) where f (x) is continuous but not differentiable.
(c) Locate the x-value(s) where f (x) is not differen- tiable.
x
y
โ 1 0 1 3 4 โ 2
โ 1
1
3
(3) 4. Find the value(s) of k such that f (x) is continuous:
f (x) =
kx^2 + 2k^2 x โ 4 if x โค 1 4 kx^2 + k^2 x + 6 if x > 1
(5) 5. Using only the limit definition of the derivative, show that if f (x) = x^2 โ 5 x + 3 then f โฒ(x) = 2x โ 5.
(28) 6. Find dy dx for each of the following functions. Do not simplify your answers. (a) y = 8 x^3 โ 5
x + 2xe+1^ + e^3
(b) y =
5 x โ 3 2 โ 7 x
(c) y = esec(5x)^ cot(7x)
(d) y = ln
2 x โ 3 cos(x) (3x^2 โ x)^3
(e) xy = (x + 3y)^4 + 7 (f) y = 4 x^ log 4 (
x) (g) y = (3x + 2)cos(x)
(4) 7. Find the second derivative of f (x) = xe^5 โ^6 x^ + xโ^1. (4) 8. Find the absolute maximum and absolute minimum of f (x) = x^3 + 2x^2 โ 15 x + 27 on the interval [โ 4 , โ2].
(4) 9. Find an equation of the line tangent to the curve of y = x^2 + 1 x โ 2 at x = 4.
(4) 10. Use the second derivative test to find the relative extrema of f (x) = x^4 โ 4 x^3 + 4x^2 + 3.
(Marks)
(6f) dy dx = 4x^ ln(4). log 4
x + (^12)
x ln(4)
. 4 x^ ; (6g) dy dx = (3x + 2)cos^ x
[
โ sin(x) ln(3x + 2) + 3 cos(x) 3 x + 2
]
(7) f โฒโฒ(x) = โ 12 e^5 โ^6 x^ + 36x e^5 โ^6 x^ + 2xโ^3 (8) absolute maximum is 63 when x = โ3 ; absolute minimum is 55 when x = โ 4 (9) y = โ^14 x + 192 ; (10) relative minimum at (0, 3) and (2, 3) ; relative maximum at (1, 4) (11a)
x โ intercepts :
y โ intercept: none vertical asymptote: x = 0 horizontal asymptote: y = 5 relative minimum: (โ 2 , 9) points of inflection:
increasing: x < โ2 or x > 0 decreasing: โ 2 < x < 0 concave up: x < โ 3 concave down: โ 3 < x < 0 or x > 0
(11b)
x
y
โ 3 โ 2 0 4
5
9 8
(12a) R(x) = 1000x โ x^2 ; (12b) C(x) = 3000 + 20x ; (12c) P (x) = โx^2 + 980x โ 3000 (12d) an increase from 300 to 301 units would result in an increase of about 380 $ in profits (12e) 490 units to maximize profit
(13) A courtyard with dimensions 90 meters by 80 meters to minimize cost
(14a) ฮท(5) = โ 5 .5 ; (14b) demand has unit elasticity at x = 10 units
