1. Let f(x) = x - 3x. a) Find f (x) and f (x). [1 Mark] 2. Differentiate the following functions. Do not simplify your answers. Each part is worth 1 Mark. a) f (x) = e (x + 2x) b) Find the critical points of f (x). [1 Mark] Inx b) g(x) = 2-3 c) Give the intervals for which f is increasing and intervals is decreasing. [1 Mark] c) h(x) = V1 - e - x d) Identify which critical points are maximum, minimum or neither. [1 Mark] d) f(x) = 6x + x - 3x + 1 e) Sketch the graph of f. [Please label the x-axis and y-axis to get full points] [1 Mark] e) f(x) = 4x2 - 2x - 73. Letf(x) = x - 2x + 1 Find f (x) using the limit definition of derivative. 4. The implicit equation 4x2 + y? = 8 defines the curve y in terms of X. [3 Marks] (i) Find dy dx [2 Marks] (ii) Find the equation of the tangent line to the graph of f at the point (2, 1) (ii) Find the equation to the curve at the point (1,2) [3 Marks] [2 Marks]5. A retail store has been selling 200 televisions weekly at $350 each. A marketing survey shows that for each $10 rebate offered to consumers, the number of units 7, A person who is 6 feet tail is walking away from a lamp post at the rate of 40 feet sold will increase by 20 weekly. per minute. When the person is 10 feet from the lamp post, his shadow is 20 feet long. Find the rate at which the length of the shadow is increasing when he is 30 :1) Find the demand function, p(x). [1 Marks] feet from the lamp post, [5 Marks] 17) Find the revenue function, R(x). [1 Marks] c) How large of a rebate should the store offer to maximize revenue? [3 Marks] 6. The area ofa circular oil spill expands the rate of 16ft3 per minute. How fast is the radius increasing when the area is 64fl3? [5 Marks] 1. A street light is mounted at the top of a 15 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 5 / sec along a straight path How fast is the tip of 3. Find the following integral by an appropriate method his shadow moving when he is 40 ft from the pole? [5 Marks] f xzexdx [5 Marks] 2' Find the area of the region enclosed by y = 2", y = x2 1, x = 1 and = 1. x [5 Marks] 4, Use logarithmic differentiation to nd if = 2x7723 (3x+1)5 [5 Marks] 6. For the function f (x) = e" 2x + -x 5. Suppose the supply function for a certain item is given by S(q) = (q + 1) and the demand function is given by D(q) = 1000 q+1 a) find the intervals on which f is increasing and decreasing. a. Find the point at which supply and demand are in equilibrium b. Find the consumer's surplus c. Find the producer's surplus b) find the local maximum and local minimum values. c) find the intervals of concavity and the point inflection points.7. Suppose $5000 is invested paying 2.5% interest per year (APR). How long will it take to have $7000 if interest is compounded continuously. [5 Marks]