In order to receive full credit all work must be shown and readable as discussed in class. Starred* problems are problems you would see on a calculator section. If the problem isn't starred you should not use a calculator. 1. *The point P(8, In (8 ))lies on the curve y = In(x). a. If Q is the point (x, In (x)), use your calculator to find the slope of the secant line PQ (correct to four decimal places) for the following values of x in Table 1 below. No need to show work here, just fill in the table. X 7.9 7.99 7.999 8.001 8.01 8.1 b. Using the results of part (a), estimate the value of the slope of the tangent line to the curve at P(8, In (8)) 2. For the curve f(x) = 5x - x' what is the slope of the secant line through the point P= (1, f(1) and Q (4, / (4) )?15. Given the function determine where the function is discontinuous. Justify each discontinuity by showing how it doesn't satisfy the definition of continuity. 2x3 -1 if x 4 Some help (not all): A function is continuous at a point a if lim f(x) = f(a) 16. Figure 3 shows the first derivative for the function f(x) over the interval [0,10]. a. Over what interval(s) is f(x) increasing? Y vs. X How do you know? b. Over what interval(s) is f(x) concave up? How do you know? c. Where does f(x) have its local maximum value? How do you know?IT. Given The graph of y: x}, sketch the graph of y =f'ix} an the same set of axes. For Figure l and Figure 2. a. HIIId In. K 18. Figure 3 shows the graph of the derivative of a function f. Use the graph to answer the following questions about the original function f. Figure 3 (a) State the interval(s) upon which f is increasing. (b) State the interval(s) upon which f is decreasing. (c) State the interval(s) upon which f is concave up. (d) State the interval(s) upon which f is concave down. (e) State the number(s) where f has a local maximum. (f) State the number(s) where f has a local minimumA projectile is launched vertically upward from the surface of Mars. Table I gives the height of the object at the indicated time following launch. Table 1 Time(seconds) 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 Height (feet) 0 18.2 34.4 48.4 60.4 70.4 78.3 84.2 88.1 90.0 Time(seconds) 4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2 7.6 Height (feet) 89.6 87.3 82.9 76.5 68.1 57.6 45.1 30.5 13.8 0 a) Using the data, compute the average velocity of the projectile on the following time intervals: i) [0. 4.0] ii) [1.6, 2.4] ii) [1.6, 2.0] iv) [2.0, 2.4] b) Estimate the velocity of the projectile when t = 2.0. Justify your results.4. "Consider the function f(x)=x - 3x+ 6 |x - 31 . Make an appropriate table of values in order to determine the indicated limits. a) lim f(x) = b) lim f(x) = X-3 X-3 x) fx) c) Does lim f(x) exist? If it does, what is it? If not, why not? x-3 5. The graph of f is given in Figure I. Use it to evaluate the limits, if they exist. If they do not exist, explain why. a.) lim (f(x)) b.) lim (f(x)) c lim (f(x)) d) lim (f(x))6. For the function f whose graph is given, state the value of the given quantity, if it exists. If it does not exist, briefly explain why. 3- A a) lim (f(1)) b) lim (f(t)) c) lim (f()) d) lim (f(1)) e) lim (f(1)) 1-2 () f (2) g) lim (f(1)) h ) f (4 )7. State each value of x where the function f shown in figure 1 is discontinuity. Explain why each value is discontinuous. Figure 1:/ *=4 8. *If a ball is thrown straight up into the air with an initial velocity of 40 ft's, then the height (ft) of the ball / seconds after it is thrown is given by the function s(1) = 401 -167'. What, including the unit, is the value of s'(2) and what does this value tell you in the context of the problem?\fsin x g.) lim= i.) lim(vx*+1-x) *-# 2+COSX j.) lim- *+x 14 3-x h.) Evaluate lim- 3.x - x-2 1 5x3 + 4x+1 k.) limx-x10.) Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown 2- 0 11. Sketch a possible function that satisfies the given conditions. Label and scale each axis. a. lim f(x)= -2 b. lim f(x)=0 c. lim f (x)=0 d. lim f (x)= -60 e. lim f (x ) =2 f. f is continuous from the right at 3.12. Use the definition of a derivative to find f'(2) where f(x) = x] -2x 13. Find the equation of the tangent line to the curve f(x) =x' -2x at the point (2, 4) 14. If f(x) =