1. Use the limits definition f' ( x ) = lim (x+h)-f(x) h-0 h to find an expression for the derivative to the functions; a. f(x) = 7x b. f(x) = c. f(x) = - 20x3 2. a. Use the limits definition f* ( x ) = lim (xth)-f(x) to find an expression for the derivative to the 1-0 h function y = =x3. b. Using your answer to a) find the slope of the tangent to the curve y = = X to complete the following table. X Slope of tangent (m = y') -3 -2 -1 0 2 3 3. a. After looking closely at the problems and solutions presented in this activity, describe the patterns that you see emerge as you find the derivatives of simple functions like y = x", y = x*, and y = 1/x. (Hint - It might be helpful to remember that, although y = 1/x is written as a rational function, it can also be expressed as a power function. That is, it can be expressed in the form y = x"'). b. Using the patterns that you described in a), find the function that would produce the following derivative of y' = X. 4. a. Find an expression for the instantaneous rate of change of distance with respect to time (velocity) of a falling object if s(t) = -4.91- where s is distance (meters) and t is time (seconds), ignore the effect of air resistance. b. Use your answer for a) to find the velocity at the times listed on the table below. From Physics, you may recall that the velocity is equal to the slope of the distance/time function. Time Velocity (s'(t) = v(t)) 10 sec 45 sec 2 min