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(Marks)
(5) 1. Use the graph of the function f (x) below to determine the following. Use ∞, −∞, or DNE where appropriate.
(a) lim x→−∞
f (x)
(b) lim x→+∞
f (x)
(c) lim x→− 3
f (x)
(d) lim x→ 2 −^
f (x)
(e) lim x→ 2
f (x)
(f) lim x→− 1
f (x)
(g) lim x→ 0 +^
f (x)
(h) f (−3)
(i) List the x-value(s) at which the func- tion f (x) is continuous but not differ- entiable.
x
y
− 3 2
− 1
1
2
3
(10) 2. Evaluate the following:
(a) lim x→ 2
x^2 + 2x − 8
x^4 − 16
(b) lim t→ π 3 −
sin t
1 − 2 cos t
(c) lim x→ 4
4 − x √ x + 5 − 3
(d) lim x→−∞
2 + 9x
5 + 4x
(e) lim x→ 2
x ln x − ln(x^2 )
x − 2
(4) 3. Given f (x) =
|x − 2 | if x ≤ 1 √ x − 1 if 1 < x < 5 2 x
10 − x
if x > 5
find all values of x where f (x) is not continuous, and justify your answers. Give the type of discontinuity at each value.
(4) 4. (a) Use the limit definition of the derivative of a function to find the derivative of f (x) =
2 x
7 − x
(b) Check your answer to part (a) by using the derivative rules.
(Marks)
(15) 5. Find
dy
dx
for each of the following:
(a) y = (3x^2 − 9) sec^4 (5x + 3)
(b) y = ln
(x^2 − 9)^4
x^3
x + 7
(c) y = sin [(2x − 1)^3 ] + cos^3 (2x − 1)
(d) y = (sin x)ln^ x
(e) y = e
√ x^2 +
(3) 6. Find an equation of the tangent line to the curve y = (1 + e −x ) 4 at x = 0.
(4) 7. Given f (x) =
x 2
(x + 1)^2
(a) Find f ′(x) and simplify.
(b) Determine whether f (x) has a relative (local) maximum or minimum at x = 1.
You may use the fact that f ′′(x) =
−4(x − 2)
(x + 1)^4
(5) 8. Given the curve x^2 − xy + y^2 = 9 ,
(a) Show that y′^ =
y − 2 x
2 y − x
(b) Find the coordinates of both points on the curve where the tangent line is horizontal.
(8) 9. Given f (x) =
x^3 − 8
x^3 + 8
and f ′(x) =
48 x^2
(x^3 + 8)^2
and f ′′(x) =
− 192 x(x^3 − 4)
(x^3 + 8)^3
Sketch the graph of f (x) clearly showing all (if any) asymptotes, intercepts, local (relative) extrema, and points of inflection.
(3) 10. Find the absolute extrema of f (x) = x^1 /^3 − x^2 /^3 on [− 1 , 1].
(3) 11. Let f (x) = x^2 ex^. Find the intervals where f (x) is increasing or decreasing. (Do not sketch the graph.)
(4) 12. Let θ (in radians) be an acute angle in a right angled triangle and let x and y respectively be the lengths of the sides adjacent to and opposite to θ. Suppose that x and y vary with time. At the instant when x = 2 and is increasing at 4 units/second, y = 2 and is decreasing at 1 unit/second. How fast is θ changing at this time?
(4) 13. Michael has 28 m of fencing to enclose two separate turtle pens. One pen will be a rectangle three times as long as it is wide, and the other pen will be a square. For the comfort of the turtles, the width of the rectangular pen should be at least 1 m and at most 3 m. Find the maximum and minimum total areas of the pens.
(4) 14. Give an expression for (or exact value of) the following limits:
(a) lim h→ 0
sin(x + h) − sin(x)
h
(b) lim x→ 0 −
|x|
x
(c) lim n→∞
∑^ n
i=
e
2 i n
2 n
(Marks)
Answers
1.(a) ∞ (b) 2 (c) 1 (d) 2 (e) DNE (f) − 1 (g) 0 (h) 3 (i) − 1
2.(a) 3 16 (b) −∞ (c) − 6 (d) 3 2 (e) ln 2
- jump: x = 1, removable: x = 5, infinite: x = 10 4.(a)
(7 − x)^2
(b)
(7 − x)^2
5.(a) 20(3x^2 − 9) sec^4 (5x + 3) tan(5x + 3) + 6x sec^4 (5x + 3) (b)
8 x
x^2 − 9
x
2(x + 7)
(c) 6(2x − 1) 2 cos ((2x − 1) 3 ) − 6 cos 2 (2x − 1) sin(2x − 1) (d) (sin x) ln x
ln x cot x +
ln(sin x)
x
(e)
xe
√ x^2 + √ x^2 + 3
- y = − 32 x + 16 7.(a)
2(x − 1)
(x + 1)^3
(b) f ′′(1) > 0 =⇒ loc. min. at x = 1
8.(a) differentiate implicitly (b) (
- and (−
- domain: {− 2 }, y-int: (0, −1) and x-int: (2, 0), HA: y = 1 and VA: x = −2, crit num: x = 0, increasing: (−∞, −2) ∪ (− 2 , 0) ∪ (0, ∞), decreasing: nowhere, concave up: (−∞, −2) ∪ (0, 3
4), concave down: (− 2 , 0) ∪ (
4 , ∞), inflection points: (0, −1) and (
1 3 )
x
y
− 1 0 1
1
2
- abs min: (− 1 , −2), abs max: (^1 8
4 ) 11. increasing: (−∞, −2) ∪ (0, ∞), decreasing: (− 2 , 0)
- −^54 rad/sec 13. max area: 28 m^2 , min area: 21 m^2
14.(a) cos x (b) − 1 (c) e^2 − 1 (d) 1 15. c =
16.(a) 10 + ln 2 (b) 1 −
3 (c) 2 3 x
5 x
7 x
7 / 2 + C
- differentiate RHS and simplify to obtain integrand 18. z = −2 sin t + 4 cos t + et^ + 4t − 10
19.(a) A ≈ 57 4 (b)^ A^ = 3 ln 9 + 8^ 20.^
13 2 21.^ f^ (x) = 4
x and a = e 2
