<p> Derivative</p><p>LIMIT DEFINITION:</p><p> f (x  h)  f (x) f '(x)  lim h0 h</p><p>1. Let f (x)  x 2  4x  3. Find f '(x).</p><p>2. A dynamite blast blows a heavy rock straight up with a launch velocity of 160ft/sec. It reaches a height s(t)  160t 16t 2 feet after t seconds. (a) Find the average speed of the rock during the first 2 seconds after launch. (b) How fast is the rock going when it is 256ft above the ground?</p><p>3. Suppose the cost, C, in thousands of dollars, of a jacket manufacturer is given by C(x)  6  2x 2 , where x is the number of jackets produced in thousands. (c) Find the average rate of change of the cost as the production-level increases from 4000 to 4100 jackets. (d) Find the instantaneous rate of change of the cost when x = 4.</p><p>[NOTE: The instantaneous rate of change of the cost is called Marginal Cost]</p><p>4. The revenue, R, in hundreds of dollars, for a kitchen sink manufacturer is given by R(x)  10x  0.01x 2 , where x is the number of sinks sold. (a) Find the average rate of change of the revenue as the sale-level increases from 100 to 110 sinks. (b) Find the instantaneous rate of change of the revenue when x = 100.</p><p>[NOTE: The instantaneous rate of change of the revenue is called Marginal Revenue] 5. The profit, P, in millions of dollars, of a truck manufacturer is given by is given by P(x)  x 2 10x 16 , where x is the number of trucks sold in thousands. (a) Find the average rate of change of the profit as the sale-level increases from 1000 to 1100 trucks. (b) Find the instantaneous rate of change of the profit when x = 1000.</p><p>[NOTE: The instantaneous rate of change of the profit is called Marginal Profit]</p><p>6. Let f (x)  x 2  3x  4</p><p>(a) Find the slope of the secant joining the points on the graph of f corresponding to x = 2 and x = 3 (b) Find the slope of the tangent to the curve at the point corresponding to x = 2. (c) Find the equation of the tangent to the curve at the point corresponding to x = 2.</p>