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The point P(4, 27) lies on the curve y = x2 + x + 7. If Q is the point (x, z2 + x + 7), find the slope of the secant line PQ for the following values of x. If x = 4.1, the slope of PQ is: and if x = 4.01, the slope of PQ is: and if x = 3.9, the slope of PQ is: = and if x = 3.99, the slope of PQ is: = Based on the above results, guess the slope of the tangent line to the curve at P(4, 27).

m s If a rock is thrown upward on an exoplanet of a nearby star with initial velocity of 25, its height in meterst seconds later is given by y = 25t - 2.05t. (a) Find the average velocity over the given time intervals: i. [2, 3] ii. 2, 2.1 iii. [2, 2.01] iv. [2, 2.001] (b) Estimate the instantaneous velocity when t = 2. (a) On the interval [2, 3] On the interval [2, 2.1] On the interval (2,2.01] On the interval [2, 2.001) (b) The instantaneous velocity at 2 seconds is

Guess the value of the limit (if it exists) by evaluating the function at the given numbers. (It is suggested that you report answers accurate to at least six decimal places.) cos(122) – cos(2x) Let f(x) = cos(122) - cos(2x) We want to find the limit lim 10 22 22 Start by calculating the values of the function for the inputs listed in this table. f(x) 0.2 0.1 0.05 0.01 0.001 0.0001 0.00001 Based on the values in this table, it appears lim cos(12x) - cos(2x) r2 = 20