Vector Calculus Questions and Answers

Find exact values for each of the following 43 sec 30 45 csc 60 44 csc 30 46 sec 60

6 Using rules to combine known integral values Activate Suppose 7 5 4 5 6 f x dx 7 5 6 f x dx 10 10f x 8 dx 7 5 4 5 5 f x dx 8 f x dx 10 9

Problems 1 6 refer to right triangle ABC with C 90 In each case use the given information to find the six trigonometric functions of A 1 b 3 c 5 2 b 5 c 13

4 pts Find the inverse of the function f x f x X 2 X S X 2 X 5 x2 x5 0 4 y

3 Find the trigonometric function values for 6x using the graph of the unit circle A tan 60 Simplify your answer B The function is undefined OA cot 6x Simplify your answer OB The function is undefined 4 1 0 2 0 1 0 0 1 1 0 Select the correct choice below and if necessary fill in the answer box to complete your choice et A O 5

nd the a period b phase shift if any and c range of the following function 1 y 2 csc 3x 1 a The period of y 2 csc 3x is Type an exact answer using as needed Use integers or fractions for any numbers in the

In calculus it is often desired to replace radial expressions with trigonometric expressions The next group of exercises illustrates this Simplify the given expression as much as possible by making the given substitution Assume 0 0 90 65 Vx 9 x 3 tan 0 66 Vx 16 x 4 tan 0

41 sin 8 c

Show that each of the following statements is an identity by transforming the left side of each one into the right side 43 cos tan 0 sin 0 45 sin sec 0 cot 0 1 47 49 csc 0 cot 0 sec 8 csc sec 0 tan 8 44 sin cot cos 0 46 cos csc tan 0 1 48 50 sec 8 tan 8 csc 8 sec 8 csc 8 cot 0

Factor 33 2

33 25 35 1 x 37 sin 0

Multiply 21 sin 4 sin 0 3

Find the arc length of the curve below on the given interval x 3 y 1 4x on 1 2 The length of the curve is Type an exact answer using radicals as needed

Find the arc length of the curve below on the given interval X 8y 2 for 3 sys5 The length of the curve is Simplify your answer

Find the area of the region enclosed by the curves y 2 sinx and y 2 sin 2x for 0sxsx The area of the region enclosed by the curves is Type an exact answer using radicals as needed y 2 sinx ly 2 sin 2x

7 The line y 3x passes through the points 0 0 and 1 3 Find its slope

The graph to the right shows the total sales in thousands of dollars for a department store during a typical 24 hour period a Calculate the rate of sales during the period between 8 A M and 10 A M b Which 2 hour interval in the day sees the highest rate of sales and what is the rate Sales in Thousands of Dollars 24 20 16 12 8 0 2 A M 6 A M 12 noon 6 P M 10

Ex 2 Graph f x x y Name the asymptotes X Domain leftmost to rightmost Range lowest to highest X Y

Find the remaining trigonometric ratios of 12 21 cos 0 and terminates in QI 22 13 23 cos 0 25 sin 0 5 13 and 0 E QIII 27 sin 0 and 0 QIV E and terminates in QI 29 tan 0 and 0 terminates in QI 4 31 tan 0 3 and 8 E QII 33 sec 0 2 and 0 terminates in QII 35 sec 0 3 and 8 E QIII 5 37 csc and terminates in QII 0 39 csc 0 1 13 5 and 0 QIII QU 24 26 28 30 32 34 s 36 S 38 c 40 c

19 If sin 0 20 If cos 0 23 cos 25 sin 1 V5 27 sin 0 Find the remaining trigonometric ratios 12 21 cos 0 and terminates in QI 13 5 13 and 8 E QII find cos 1 1 V10 2 and E QIII fine and terminates in QI a and 8 E QIII a and 8 E QIV 3 29 tan 0 and 0 terminates in Ol

9 sin 5 13 and cos

Use the reciprocal identities in the following problems 1 If sin 0 find csc 0 2 If cos 0 V3 2 find sec 0

4 r x 3x4 9x 2

If is an angle in the first quadrant and sin 0 0 2 V find tan 0 2

3 Find the missing lengths of the sides of the triangle 2 45

Consider the following x y 3 y 4 2 2 2 4 6 8 Use the Vertical Line Test to determine whether y is a function of x O Yes y is a function of x No y is not a function of x X Solve for y to describe how you can use a graphing utility to produce the given

Assume that the domain of f is the set A 2 1 0 1 2 Determine the set of ordered pairs representing the function f f x x 5 f 2 x y f 1 x y f 0 f 1 f 2 x y x y x y

One cubic centimeter of a typical cumulus cloud contains 380 water drops which have a typical radius of 10 pm a How many cubic meters of water are in a cylindrical cumplus cloud of height 3 0 km and radius 1 0 km b How many 1 liter pop bottles would that water fill c Water has a density of 1000 kg m How much mass does the water in the cloud have a Number i b Number i c Number Units Units Units

m 6 5 8 12 Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points Use a square setting 5 y 5 10 15 20 y 20 15 10 5 5 5 10 10 X 15 15 X O 15 15 10 10 5 5 y 20 15 10 5 y 5 10 15 20 5 X

Determine the slope m and y intercept if possible of the linear equation If the slope is undefined enter UNDEFINED Enter NONE if there is no y intercept 2x 9y 1 slope m y intercept x y Describe its graph O The line passes through 0 1 and falls 3 3 units for each horizontal increase of units 9 The line passes through 0 1 and falls 2 units for each horizontal increase of 9 units O The line is vertical and passes through 2 0 O The line is horizontal and passes through 0 1

Ose the graph of f x log 4x to graph the function f x log 4 x 1 3 Use the graphing tool to graph the function Click to enlarge graph Ay

If fog x h x such that x 4x 8 which of the following could accurately represent f and g Of x 2x 4 and g x 2x 4 Of x 4x 8 and g x x Of x x and g x 4x 8 Of x 2x 4 and g x 2

Find the horizontal asymptote y 2 2 x 2x 3 U y X 2

Determine whether the following statements are true and give an explanation or counterexample a A pyramid is a solid of revolution b The volume of a hemisphere can be computed using the disk method TR 2 2 Let R be the region bounded by y sin x and the x axis on 0 The volumes of the solids generated when R and R are c Let R be the region bounded by y cos x and the x axis on revolved about the x axis are equal a Choose the correct answer below C OA The statement is false A solid of revolution must have cross sections which are circular i e either disks or washers with the axis of revolution running through the center of the disk A pyramid has circular cross sections so it cannot be a solid of revolution B The statement is false A solid of revolution must have cross sections which are circular i e either disks or washers with the axis of revolution running through the center of the disk A pyramid does not have any circular cross sections so it cannot be a solid of revolution OC The statement is true A solid of revolution must have cross sections which are circular i e either disks or washers with the axis of revolution running through the center of the disk A pyramid has circular cross sections so it cannot be a solid of revolution D The statement s true A solid of revolution must have cross sections which are circular i e either disks or washers with the axis of revolution running through the center of the disk A pyramid does not have any circular cross sections so it cannot be a solid of revolution The cross section of the hemisphere plotted in the xy plane will be a circle with a radius of 2r where r is the radius of the hemisphere The equation of this curve is The area of the disk is then z x b The statement is x y r which can be solved for y y 4 The radius of the disk passing through the point x on the x axis is y x Integrating the differential volume dV 4 x dx from gives the formula

E

19 Find the distance between the point 6 1 4 and the line x 4 3 y 5 z 1 1 1

A farmer has 1500 available to build an E shaped fence along a straight river so as to create two identical pastures The materials for the side parallel to the river cost 6 per foot and the materials for the three sections perpendicular to the river cost 5 per foot Find the dimensions for which the total area is as large as possible Dimensions include help units in your answers

K The graph to the right shows the closing share prices in dollars for Company A and Company B on the 21 trading days of the month of January 2018 What was the highest closing share price for Company A and on what day did this occur CODE The highest closing share price for Company A was Round to the nearest ten cents as needed on day COD A Price 25 24 23 22 21 20 19 18 17 16 15 14 13 M Company A Q 25 24 Q 23 Serdery B 22 21 20 19 18 17 16 15 14 Day

Decide whether each solid is a prism pyramid or neither a b c d O O O prism pyramid Oneither prism LARMPMT1 10 4 006 pyramid neither prism pyramid neither Oprism pyramid neither

List all the x intercepts Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The graph has two x intercepts The leftmost x intercept is rightmost x intercept is Type integers or decimals OB The graph has one x intercept Type an integer or a decimal OC There are no x intercepts and the 0 8 1 Ay

10 Provide your answer below ag 54 Co 2 H 32 6 69 f 2 62 1 g z 1 12 71 h z 4 12 12 h z 12 15 Evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles Enter the sum of all shaded areas in terms of t

Assume that the domain of f is the set A 2 1 0 1 23 Determine the set of ordered pairs representing the function f f x x 7 f 2 x y f 1 f 0 f 1 f 2 x y C x y x y x y

Directions Consider the following function f a at a 2 Part 1 Difference Quotient If h0 then the difference quotient can be simplified into the form 2 A f 2 h where A B and C are all non negative constants Find these constants A 2 and B49 and C 14 Note It s possible for one or more of these constants to be 0 1 f x 2x 3 2 f 2 49 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 B Ch f 2 49 B Ch f 2 lim A A 2 and B 49 and C 14 Note It s possible for one or more of these constants to be 0 That is note the negative in front f 2 h f 2 h Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 f 2 lim h0 Part 3 Tangent Line Find the equation of the tangent line to the curve y f a at a 2 Tangent Line y f 2 h f 2 h

Use the following figure and any appropriate relationship s among circular functions to 9x find tan 4 4 1 0 10 4 1 1 43 WIM 4 G 0 1 1 0 X 4 1 5 4 4 9x tan 4 Type an exact answer using radicals as needed

Use the x y coordinates in the figure to find the value of the trigonometric function at the indicated real number s or state that the expression is undefined COS 53 4 1 3x 5x 4 6 1 0 7x 6 5x 1 4 1 12 47 on 14 0 1 4x 3 13 7 2 Ay IN 3 5x 3 3x 2 0 1 x 1 AI x 4 7x 6 12 N 6 0 11x 1 0 13 2 4 Select the correct choice below and if necessary fill in the answer box to complete your choice O A 5x 3 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression COS OB The expression is undefined

Use the x y coordinates in the figure to find the value of the trigonometric function at the indicated real number s or state that the expression is undefined 7x 6 sin 12 12 1 7 on 7 0 1 2 2 3x 5x 4 6 1 0 7x 4 1 65x 4 4 4 2x 4x 3 k K 2 3x KIM K 4 5x 3 1 2 0 1 7x6 4 6 0 11x 12 12 712 2 1 0 13 12 4 4 4 OU HULING Select the correct choice below and if necessary fill in the answer bc complete your choice OA 7x sin 6 Simplify your answer including any radicals Use integers or for any numbers in the expression OB The expression is undefined

Directions Consider the following function f a at a 2 Part 1 Difference Quotient If h0 then the difference quotient can be simplified into the form 2 A f 2 h where A B and C are all non negative constants Find these constants A 2 and B49 and C 14 Note It s possible for one or more of these constants to be 0 1 f x 2x 3 2 f 2 49 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 B Ch f 2 49 B Ch f 2 lim A A 2 and B 49 and C 14 Note It s possible for one or more of these constants to be 0 That is note the negative in front f 2 h f 2 h Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 f 2 lim h0 Part 3 Tangent Line Find the equation of the tangent line to the curve y f a at a 2 Tangent Line y f 2 h f 2 h

nd the trigonometric function values for 6x using the graph of the unit circle elect the correct choice below and if necessary fill in the answer box to complete your choice A sin 6 Simplify your answer fined 1 0 2 0 1 10 0 1 1 0

Use a calculator to approximate the following sin 6 3812 cos 6 3812

K Find the length to three significant digits of the arc intercepted by a central angle 0 in a circle of radius r 2x 7 r 14 1 cm 0 radians The length of the intercepted arc is approximately cm Round to one decimal place as needed