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(Marks)
(15) 1. Use algebraic techniques to evaluate the following limits. If a limit fails to exist, use one of the symbols −∞ or ∞ as appropriate.
(a) lim x→ 10 +
x + 5
−x + 10
(b) lim x→− 2
x^3 + 3x^2 + 2x
x^2 − x − 6
(c) Find lim x→+∞
(3 + 7x)(1 − 2 x)
4 x^4 + 1
(d) lim x→ 0
sin x
sin x
x
− cot x
(e) lim x→ 1
x − 1 √ x + 3 − 2
(4) 2. Given the function f defined by f (x) =
x + 5
x^2 + 2x − 15
(a) Find both the values of x where f (x) is discontinuous
(b) Find the limit of f (x) as x approaches each of the values found in part (a)
(3) 3. Find constants a such that the function is continuous for all real numbers
f (x) =
12 x ≤ − 3
ax + 3 − 3 < x < 5
− 12 x ≥ 5
- Complete each part below
(1) (a) State the limit definition of the derivative of a function f (x).
(4) (b) Use the limit definition of the derivative to find f ′(x) for f (x) =
8 x + 17
(28) 5. Find
dy
dx
for each of the following functions. Do not simplify your answer.
(a) y =
3 x
x^2
(b) y = 3
3 x + 2
5 x^2 − 1
(c) y = 3 (sin x) 2 x
(d) y = log(x + 1) + x 3 3 x
(e) y = ln
[√
x^2 + 1 (2x + 1) 3
3
3 x^4 − 2
]
(Hint: Use the properties of logarithmic functions to simplify the problem first)
(f) xy^2 = exy^ − 3 ex
(g) y =
e^3 −x^
x + 1
cos 2x
(Marks)
(5) 6. Let f (x) = x 3 (3x + 4)
2
Find the x-coordinates, if any, at which the graph of f (x) has a horizontal tangent.
(5) 7. Find the equation of the tangent line to the graph of f (x) =
x
5 x + 1
at point
1 2
(4) 8. Use the second derivative test to find the relative (local) extrema of f (x) = 1 2 x
4 − 4 x 2
(4) 9. Find the absolute extrema of f (x) = 2x^4 − 36 x^2 + 20 on the interval [− 4 , −1].
(11) 10. Given the function f (x) = x^5 − 5 x^4 List all x and y intercepts, vertical and horizontal asymptotes, relative extrema, points of inflection, intervals where f (x) is increasing, decreasing, concave up and concave down. Use all the above and sketch a carefully labelled graph of f (x)
(5) 11. Mary has 1800 m of fence which will be used to enclose 3 sides of a rectangular field. The fourth side has a river and no fence is needed. What dimensions will give her maximum area?
(5) 12. Suppose the average cost is c = 100 + 3x + 0. 1 x^2 and the demand is p = 30x − 0. 9 x^2
(a) Find the Profit function
(b) Find the marginal profit
(c) Evaluate the marginal profit when x = 3. Interpret the result.
(6) 13. The demand function for a certain product is p =
16 − x where p is the price per unit of the product in dollars and x is the number of units of the product.
(a) State the domain of the function
(b) Find the price elasticity of demand, η
(c) State the intervals where the function is elastic, inelastic and of unit elasticity
(d) Find the price elasticity of demand when x = 9
(e) At x = 9, if the price increased by 4% what is the change in demand?
