Calculus Questions

2 Let y f x be the continuous function that satisfies the equation x 5x y 4y 0 and whose graph contains the points 2 1 and 2 2 Let be the line tangent to the graph of fat x 2 a Find an expression for y b Write an equation for line 1

3 Given the curve x xy y 9 a Write a general expression for the slope of the curve

1 Given the curve x xy 2y 6 a Find an expression for the slope of the curve at any point x y on the curve

4 Let f x 3x 2 a Determine algebraically whether the function is even odd or neither 31 1 bus S 81 olev giwollel b Find the difference quotient f x h f x wolod bobryong bing ads sall biamo 0

Find f Jes 2x Cos Cx

10 Represent f x 1 2x as a power series What is its radius of convergence

f x x ax 15 x2 9 lim f x X 3 Find the limit

The total accumulated costs C t and revenues R t in thousands of dollars respectively for a photocopying machine satisfy 1 C t 58 and R t 7t8 e t 56 where t is the time in years Find the useful life of the machine to the nearest year What is the total profit accumulated during the useful life of the machine The useful life of the machine is year s Round to the nearest year as needed Using the useful life of the machine rounded to the nearest year the total profit accumulated during the useful life of the machine is Round to the nearest dollar as needed

A person wants to purchase both parcels of land shown in the figure to the right and make a check on their combined area There is no equation for the river frontage so the person uses the average of the left sum of rectangles covering the area The 1 000 foot baseline is divided into 10 equal parts At the end of each subinterval a measurement is made from the baseline to the river and the results are tabulated Let x be the distance from the left end of the baseline and let h x be the distance from the baseline to the river at x Use L 0 to estimate the combined area of both parcels and calculate an error bound for this estimate How many subdivisions of the baseline would be required so that the error incurred in using L would not exceed 2 500 square feet X 0 100 200 300 400 500 600 700 800 900 1 000 h x 0 196 244 254 304 344 413 474 274 527 550 River ft2 Parcel 1 1 000 ft The combined area of both parcels is The error bound for the estimate is ft The number of subdivisions required so that the error incurred in using L would not exceed 2 500 square feet isn Parcel 2 h x 550 ft

The marginal average cost of producing x digital sports watches is given by the function c x where C x is the average cost in dollars c x 1 400 2 Find the average cost function and the cost function What are the fixed costs C 100 25 The average cost function is C x The cost function is C x 0 The fixed costs are 1400

Calculate the definite integral by referring to the figure with the indicated areas Area C 5 856 Area D 1 71 Area A 1 549 Area B 2 734 to

Calculate the indicated Riemann sum S4 for the function f x 34 4x Partition 0 12 into four subintervals of equal length and for each subinterval X 1X let c 2xx 1 xx 3 SA Simplify your answer SKELB

Find the particular solution of the differential equation dy 5 dx 1 x 1 x y 0 9

39 40 Find the vectors T N and B at the given point 39 r t 1 1 1 1 3 1

13 16 a Find the unit tangent and unit normal vectors T t and N t b Use Formula 9 to find the curvature 13 r t t 3 cos t 3 sin t

41 42 Find equations of the normal plane and osculating plane of the curve at the given point BO smo ob impu 41 x 2 sin 3t y t z 2 cos 3t 0 2

1 4 Find the length of the curve 1 r t t 3 cos t 3 sin t 5 t 5 2 r t cos ti sin tj In cos tk 0 t T 4 3 r t i t j t k 0 t 1 3

Refer to the graph to the right and follow the directions in parts a through f a Is a 1 or is 0 a 1 b Give the domain and range of f and the equation of the asymptote c Sketch the graph of g x a d Give the domain and range of g and the equation of the asymptote e Sketch the graph of h x a f Give the domain and range of h and the equation of the asymptote a Is a 1 or is 0 a 1 Choose the correct answer below 1 a y f x a 1 a 0 1 O OA For the function f x a 0 a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis OB For the function f x a a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis OC For the function f x ax a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis OD For the function f x a 0 a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis b Give the domain and range of f and the equation of the asymptote The domain of f is

1 Find the Taylor series for f x sin x centered at a

sec x y 6 sec ase AWY a The period is Type an exact answer in terms of b Select the correct choice below and fill in the answer box within your choice OA The phase shift is B The given function has no phase shift c Find the range of the function Choose the correct answer below O A 00 00 OC 6 6 units Type an exact answer in terms of OB 6 U 6 0 OD 6 U 6 00

e6x 5x e OA 0 924 OB 0 636 OC 0 778 O D 2 196

10 O A 10 2 8 10 10 2 8 5 8 1 8 10 10 Q B 10 1 8 3 0 10 10 S O C 10 10 2 5 0 3 10 O D 10 10 2 5 0 3

16 0 16 Type an integer or a simplifed fraction

sec tan OA 3 OB 2 O C O D 1 5 3 2 2 3 3

K Solve the equation for solutions in the interval 0 2 3 sec2x 9 secx 2 4 A OB O C OD 2n 4x T 3 3 5t 7x IC 6 6 7x 3x 11x 62 6 2x 4x 3x 3 3 2

Refer to the graph to the right and follow the directions in parts a through f a Is a 1 or is 0 a 1 b Give the domain and range of f and the equation of the asymptote c Sketch the graph of g x ax d Give the domain and range of g and the equation of the asymptote e Sketch the graph of h x ax f Give the domain and range of h and the equation of the asymptote a Is a 1 or is 0 a 1 Choose the correct answer below 1 a f x ax 1 a 0 1 b Give the domain and range of f and the equation of the asymptoto OA For the function f x ax 0 a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis X OC For the function f x a a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis OB For the function f x a a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis OD For the function f x a 0 a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis Q G Ly

Choose the equation that matches the rational function graphed 10 8 6 4 6 X 8 10 5 OA OB O C O D f x f x f x f x x x x 6 x x x 6 x x x 6 x x x 6

Solve the equation and express the solution in exact form In ex In e In e9 OA 18 OB 7 O C 11 OD 7

f x OA 2 if x 1 4 xif x 1 5 B OC OD G

Use the graphs to evaluate gof 0 y f x 2 2 6 3 2 2 6 3 3 1 1 1 123 0 2 OA 3 OB 5 O C 6 6 X y g x 34 6 3 1 2 3 2 4 2 4 1 2 3 6 X Q

tan 13x 12 OA 2 3 2 3 4 OB 2 3 4 On 2 3 O C

cos x 2 sin x cos x secx O A cot x OB cScx OC sec x OD csc 2x

Determine the domain of the function f x 49 36x OA OB O C 77 6 6 0 6 17 00 OD

3 For the following differential equation 27 y ry x 5 y 0 do the following a Show that r 0 is a regular singular point b Find the indicial equation and the indicial roots of it c Use the Frobenius method to find two series solutions of the equation Show the full explanation for this problem

Compute the surface area of revolution about the x axis over the interval y x ln x 1 e

18 tan 0 36 cos 330

15 cos 0 60

17 sin 0 35 300

8 tan 0 330

16 tan 120

7 sin 0 300

14 tan 0 600 y

300

cos 609

0 cos 0 120

Find the exact length of the polar curve 82 0 0 3 t

60 y sin 0 90 180 270 360 450 5405

2 u 45 V

0 10 60 y

y 4 6 45 45 4 6