WeBWorK demonstration assignment

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1 WeBWorK demonstration assignment.( pt) Match the statements defined below with the letters labeling their equivalent expressions. You must get all of the answers correct to receive credit.. x is less than 5. The distance from x to 5 is less than or equal to. x is less than or equal to 5. x is greater than or equal to 5 5. The distance from x to 5 is more than A. x 5 B. x 5 C. x 5 D. x 5 E. x 5.( pt) The inequality x 5 x 0 x means that x is in the closed interval A B where A is: and B is:.( pt) Solve the inequality x 5x 6 0. The solution is x is in the open interval A B where A is: and B is:.( pt) Consider the inequality x x 5 The solution of this inequality consists one or more of the following intervals: A, A B, and B where A B. Find A Find B For each interval, answer YES or NO to whether the interval is included in the solution. A A B B 5.( pt) To say that x 8 is the same as saying x is in the closed interval A B where A is: and where B is: 6.( pt) To say that x 5 5 is the same as saying x is in the closed interval A B where A is: and where B is: 7.( pt) The equation of the line that goes through the point and is parallel to the line 5x y can be written in the form y mx b where m is: and where b is: 8.( pt) The equation of the line that goes through the point 6 and is perpendicular to the line 5x y can be written in the form y mx b where m is: and where b is: 9.( pt) For each of the followings angles (in radian measure), find the cos of the angle (your answer cannot contain trig functions, it must be an arithmetic expression or number): π 6 π π π π π 0.( pt) If sin θ cos θ equals tan θ equals sec θ equals, 0 θ π, then

2 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

3 WeBWorK demonstration assignment.( pt) Let F be the function below. If you are having a hard time seeing the picture clearly, click on the picture. It will expand to a larger picture on its own page so that you can inspect it more clearly. Evaluate each of the following expressions. Note: Enter DNE if the limit does not exist or is not defined. a) lim F x = x b) lim F x = x c) lim x F x = d) F = e) lim F x = x f) lim F x = x g) lim x F x = h) lim x F x = i) F =.( pt) f(x) g(x) The graphs of f and g are given above. Use them to evaluate each quantity below. Write DNE if the limit or value does not exist (or if it s infinity).. lim g x x. f g. lim g x x. lim f g x x.( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false!. For reference you can find some definitions here. You must get all of the answers correct to receive credit.. Every bounded sequence has an accumulation point.. Every bounded sequence has a subsequence which converges to a limit point.. The sequence of rational numbers.,.,.,.59,... which approximates the

4 ratio of the circumference of a circle and its diameter, has a rational number as its limit point.. Every sequence which is convergent must be bounded..( pt) If the tangent line to y at (6, -7) passes through the point (, 0), find A. f 6 B. f! 6 5.( pt) The point P 0 lies on the curve y x x 8. If Q is the point x x x 8, find the slope of the secant line PQ for the following values of x. If x ", the slope of PQ is: and if x " 0, the slope of PQ is: and if x " 9, the slope of PQ is: and if x " 99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P 0. 6.( pt) Evaluate the limit lim t 5 t 5 t 5 7.( pt) Evaluate the limit lim x x x Determine the limits for the function f at " 87. lim = x # 87 f " 87 = lim = x # 87 Is this function continuous at " 87?: (Y or N) Can this function be made continuous by changing its value at " 87?: (Y or N) 9.( pt) a - 0 lim DNE -0 x a lim DNE x a f a 0 lim g x DNE x a lim g x 0-0 DNE x a g a Using the table above calcuate the limits below. Enter DNE if the limit doesn t exist OR if limit can t be determined from the information given.. f g. lim f g x x. lim g x x. f $ g 0.( pt) Evaluate lim x x 5 5x %" Enter the letters corresponding to the Limit Laws that you used to find this limit: 8.( pt) Limit Laws A. Sum Law B. Quotient Law C. Root Law D. Difference Law E. Constant Multiple Law F. Product Law G. Power Law Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

5 WeBWorK demonstration assignment.( pt) If x 8x, find f & x. Find f!..( pt) If 6x 8 6x 5 6x 5x, find f! x. Find f..( pt) If f t' t t 5t, find f t. Find f!..( pt) Let f t( t 7t ) 7t. (a) f! t* (b) f! [NOTE: Your answer to part (a) should be a function in terms of the variable t and not a number! Your answer to part (b) should be a number.] 5.( pt) If f t 5t, find f! t. Find f!. 6.( pt) If find. Find f!. 7.( pt) If find. Find f. + x 8 x 6 x 5x, x 8.( pt) If 6 5 x 5 x, find f! x. Find f!. 9.( pt) If 7x, x x -, find. x Find f. 0.( pt) If + 5x 5 x. x x, find. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

6 WeBWorK demonstration assignment.( pt) Let f! +0/ 5x 5x 8.( pt) If sin x, find f! x. Find f!..( pt) If sin x, find f! x. Find f!..( pt) Let sec 6x 5.( pt) If sin sin x, find f! x. Find f!. 6.( pt) Let 7sin x 7.( pt) If 6x arctan 9x, find. 8.( pt) If 8 sin x arcsin x, find f! x. 9.( pt) If 8arctan 8x, find. Find f!. 0.( pt) Let Then dy dx = y tan / x Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

7 WeBWorK demonstration assignment.( pt) The function x x 60x has one local minimum and one local maximum. This function has a local minimum at x equals with value and a local maximum at x equals with value.( pt) The function x x has one local minimum and one local maximum. This function has a local minimum at x equals with value and a local maximum at x equals with value.( pt) Consider the function ' x x 5. is increasing on the interval A and decreasing on the interval A where A is the critical number. Find A At x A, does have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER..( pt) Consider the function x 5 5x 0x. For this function there are four important intervals: A, A B, B C, and C where A, B, and C are the critical numbers. Find A and B and C At each critical number A, B, and C does have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER. At A At B At C 5.( pt) For x 56 7 the function f is defined by x 5 x On which two intervals is the function increasing (enter intervals in ascending order)? to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 6.( pt) Answer the following questions for the function x/ x 6 defined on the interval 7 6. A. is concave down on the region to B. is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The maximum for this function occurs at 7.( pt) Answer the following questions for the function x x 6 defined on the interval Enter points, such as inflection points in ascending order, i.e. smallest x values first. Enter intervals in ascending order also. A. The function has vertical asympototes at and B. is concave up on the region to and to C. The inflection points for this function are, and

8 8.( pt) Consider the function 8 x. 6 5x.. For this function there are two important intervals: A and A where the function is not defined at A. Find A For each of the following intervals, tell whether is increasing (type in INC) or decreasing (type in DEC). A : A Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether is concave up (type in CU) or concave down (type in CD). A : A 9.( pt) A fence feet tall runs parallel to a tall building at a distance of feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 0.( pt) Let Q 9 0 and R : 8 be given points in the plane. We want to find the point P x 0 on the x-axis such that the sum of distances PQ PR is as small as possible. (Before proceeding with this problem, draw a picture!) To solve this problem, we need to minimize the following function of x: + over the closed interval a b where a and b. We find that has only one critical number in the interval at x where has value Since this is smaller than the values of at the two endpoints, we conclude that this is the minimal sum of distances. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

9 WeBWorK demonstration assignment.( pt) You are given the four points in the plane A ;, B < 8= 6, C < 0, and D. The graph of the function consists of the three line segments AB, BC and CD. Find the integral dx by interpreting the integral in terms of sums and/or differences of areas of elementary figures. dx.( pt) Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. 6 6 / 6 x dx.( pt) Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. 0 6x dx.( pt) Consider the integral 7 x x dx (a) Find the Riemann sum for this integral using right endpoints and n. (b) Find the Riemann sum for this same integral, using left endpoints and n 5.( pt) Consider the integral 8? (a) Find the Riemann sum for this integral using right endpoints and n. (b) Find the Riemann sum for this same integral, using left endpoints and n dx 6.( pt) Consider the function ' x 5. In this problem you will calculate by using the definition a b dx 0 n lim n A i xc ib x 5 dx The summation inside the brackets is R n which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. x Calculate R n for DE 5 on the interval 0 and write your answer as a function of n without any summation signs. You will need the summation formulas on page 8 of your textbook (page 6 in older texts). R n lim n R n 7.( pt) The following sum n F n 6 n F n 9 n F n G""" n n F is a right Riemann sum for a certain definite integral b dx n

10 H H H using a partition of the interval b into n subintervals of equal length. Then the upper limit of integration must be: b = and the integrand must be the function = 8.( pt) The following sum 6 n F? 6 8 n F? Ï"""! 6 n n F? is a right Riemann sum for the definite integral b dx where b = and = It is also a Riemann sum for the definite integral 6 c g x dx where c = and g x = The limit of these Riemann sums as n J is Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR