Definite Integrals Questions and Answers

35 The graph of f is shown Evaluate each integral by inter preting it in terms of areas a f f x dx c ff x dx e ff x dx YA 2 0 2 b ff x dx d ff x dx f f f x dx y f x 4 6 8 AX x

27 34 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral 28 f x dx 27 f 3x dx

19 22 Express the limit as a definite integral on the given interval exi n11 Xi 19 lim n Ax 0 1 20 lim x 1 1x Ax 2 5

ing delmes a one to one function find its inverse If not write Not one to one 3 3 4 3 5 7 6 5 OA Not one to one OB 3 3 5 7 5 4 OC 3 5 3 7 5 4 OD 3 3 4 7 5 5

Consider the equation below If an answer does not exist enter DNE f x x4 8x2 7 a Find the interval on which f is increasing Enter your answer using interval notation X Find the interval on which f is decreasing Enter your answer using interval notation b Find the local minimum and maximum valu s of f local minimum value local maximum value c Find the inflection points x y x y smaller x value larger x value Find the interval on which f is concave up Enter your answer using interval notation Find the interval on which f is concave down Enter your answer using interval notation

Find the limit Use l Hospital s Rule where appropriate If there is a more elementary method consider using it x2x lim x 0 2x 1 1 log 2 X

Find the limit Use l Hospital s Rule appropriate If there is a more elementary method consider using it 18 3 In x 3x X 1 X 1

8 T F with justification The graph of the semicircle on 1 1 below fits the hypotheses of the Mean Value Theorem 1 Y 1 X

4 T F with justification If f x is a differentiable function on a b and f x has a local maximum or minimum value at x c in a b then f c 0 I 5 T F with justification If f x is a differentiable function on a b and f c 0 for a number c in a b then f x has a local maximum or minimum value at x C

Write name date and period Explain all answers including Multiple Choice Each problem is worth 4 points 1 The number of shoppers in a shopping mall t hours after it opens is given by S t The number of store cle needed to attend to s shoppers is given by C s Interpret the statement C S 3 11 in the context of this problem

204 Find dy dr if y e6x 2r

4 21584936490539 The answer above is NOT correct Find the area inside one leaf of the rose 1015 T 1 sin 50 240

Find the area inside one leaf of the rose The area is leulator Turn in a sketch of the curve r 1 sin 50

Find the area inside one leaf of the rose The area is Turn in a sketch of the curve r 1 sin 50

2 Three step Adams Bashforth method Consider the interpolation nodes t 2 1 t where t t j jh a Write down the Lagrange interpolation polynomial P t for the function f t y t at the nodes above b Consider the IVP y t f t y t integrate the Lagrange polynomial P t from a exactly to determine bi b and b3 such that Yi 1 yih b f ti Yi b f ti 1 Yi 1 b3f t 2 2 As usual y denotes the value of y t at t tk Your answers b should be given as fractions Hint you may note for b we have f L t dt 2h evaluate the integral and do the same for b2 b hb pti 1 S t ti 1 t t 2 dt

QUESTION 1 Decide whether or not the functions are inverses of each other f x 7x 4 g x Yes No 7

Rewrite the quadratic function f x 5x 10x 9 in standard form f x a x h k and give the vertex help numbers f x Vertex help points

In 2000 the birthrate in a certain area increased from 16e0 02t million births per year to 21e0 02t million births per year where it is the number of years since 2000 y 60 50 40 Birthrate 30 20 10 population change 21 e0 02t 16 e0 02t 10 20 30 40 50 Years Find the total increase in population in millions that will result from this higher birth rate between 2000 t 0 and 2050 t 50 Round your answer to the nearest whole number million

Prove the identity tan x tan x Use the Subtraction Formula for Tangent and then simplify tan x tan tan x 1 tan x x tan x 0 1 tan x

Find the area between the curves y ex and y e x shown below from x 0 to x 1 Leave the answer in its exact form y 21 18 15 12 9 6 3 0 5 y e x y e

Verify the identity sin w sin w cos w tan w 1 tan w

The displacement from equilibrium of an oscillating weight suspended by a spring is given by Y t 1 cos 78 where y is the displacement in feet and it is the time in seconds see figure Find the displacement when tw0 t and t Round your answers to four decimal places 0 2 0 y Equilibrium 6 t 1 y Y O ft ft 0 ft At In 11 Maximum negative Maximum positive displacement displacement Simple Harmonic Motion Oft

The following parametric equations define an ellipse Al WWW Duran x 4 cos 0 Answer Compute the area enclosed by this ellipse Hint The ellipse is symmetric over the x axis Select limits for integration so that you are integrating left to right y 2 sin 0 0 0 2TT Graph the curve on a calculator Turn in a sketch of the curve

Let f x f x 2x 7 7x 8 Find f x help formulas

8 5 points Let f t describe the temperature in degrees Fahrenheit of a cup of coffee t minutes after it has been brewed a 2 points What are the units of f t b 3 points Give an interpretation of the following mathematical expressions in the context of this problem for a non mathematical friend i f 0 200 ii f 0 10 iii f 0 0

1 Use the Ratio Test to determine if 2n 1 n3 1 converges or diverges

The price of a condominium is 109 000 The bank requires a 5 down payment and one point at the time of closing The cost of the condominium is financed with a 30 year fixed rate mortgage at 7 Use the following formula to determine the regular payment amount Complete parts a through e below PMT P nt a Find the required down payment b Find the amount of the mortgage c How much must be paid for the one point at closing his Round to the nearest dollar as needed d Find the monthly payment excluding escrowed taxes and insurance Round to the nearest dollar as needed e Find the total cost of interest over 30 years Dound to the nearest dollar as needed

34 sin 450 Convert each degree measure into radians 35 300 37 100 Convert each radian measure into degrees 36 930 201

A graphing calculator is recommended Use a graphing calculator to evaluate the definite integral rounding your answer to three decimal places Hint Use a command like FnInt or ff x dx 471 Lets d dx X

5 f r x f x dx for the function f x graphed below Hint No calculus is necessary just as we used integrals to find areas we can use areas to find integrals The curve shown is a quarter circle Find 53 y 24 1 2 square units 3 4 5

4Listen If the point 3 5 of Cost and tant a cost b cost 05 20 O d 5 is the terminal point of t on the unit circle tant tant cost 3 4 3 Od cost 1 tant 4130 31 10 tant 3

Givensin e cot 0 0 3 103 00 3 v3 3 CY 0 2 3 for an angle in quadrant IV find the exact value of

Which angle below is NOT co terminal with the angle O a Ob 13m 4 O c 4 3n Od 21 4

To find the distance across a canyon from point B to point C a surveying team locates points A and B on one side of the canyon and point C on the other side of the canyon The distance between A and B is 93 yards The angle CAB is 67 and the angle CBA is 89 Find the distance across the canyon from point B to point C round to the nearest yard 230 yards 150 yards 210 yards 350 yards

If the graph of y ax b x c x 3 then a c A 5 B 1 1 has a horizontal asymptote y 2 and a vertical asymptote C 0 D 1 E 5

Express the repeating decimal as the ratio of two integers 0 17 0 171717 The ratio of two integers is Type an integer or a simplified fraction

If the point P 20 of CSc t O a 20 21 Ob 21 20 c 29 21 la 21 29 29 d 29 20 is the terminal point of t on the unit circle find the v

Question 5 10 points 5 A rectangular hole is to be cut in a wall for a vent If the perimeter of the hole is 48 cm and the length of the diagonal is a minimum what are the dimensions of the hole Paragraph v BI U A v 11 OB o v X

Question 8 10 points 8 A race track 400 m long is to be built around an area that is a rectangle with a semicircle at each end Find the open side of the rectangle if the area of the rectangle is to be a maximum Paragraph B I U A lih

41 Height estimate The distance an object falls when released from rest under the influence of Earth s gravity and with no air resis tance is given by d t 16r where d is measured in feet and t is measured in seconds A rock climber sits on a ledge on a verti cal wall and carefully observes the time it takes for a small stone to fall from the ledge to the ground a Compute d t What units are associated with the derivative and what does it measure b If it takes 6 s for a stone to fall to the ground how high is the ledge How fast is the stone moving when it strikes the ground

Let h x fog x x 1 4 Find f x given g x x 1 f x help formulas

On a hot summer afternoon a city s electricity consumption is 3t2 18t 10 units per hour where it is the number of hours after noon 0 ts 6 Find the total consumption of electricity between the hours of 1 p m and 6 p m units

A graphing calculator is recommended For the function do the following f x 32 3x from x 1 to x 2 a Integrate by hand to find the area under the curve between the given x values square units b Verify your answer to part a by having your calculator graph the function and find the area using a comman Fnint or f x dx square units

From the Table 5 2 we deduce that the value of f x may be made smaller than any given number by choosing a negative real number very close to 0 In symbols we write lim f x 0 x 0 to be read as the left hand limit of f x at 0 is minus infinity Again we wish to emphasise that is NOT a real number and hence the left hand limit of fat 0 does not exist as a real number The graph of the reciprocal function given in Fig 5 3 is a geometric representation of the above mentioned facts 110 MATHEMATICS Rationalised 2023 24 f x lim f x lim x 2 c 2 f c X C X C Thus fis continuous at all points x 1 Case 3 If c 1 then the left hand limit of fat x 1 is Example 10 Discuss the continuity of the function f defined by x 2 if x 1 x 2 if x 1 lim f x lim x 2 1 2 3 x l x l The right hand limit of fat x 1 is f x 2 2 lim f x lim x 2 1 2 1 x 1 x 1 1 1 2 2 lim f x lim x 2 1 2 3 x 1 The right hand limit of fat x 1 is lim f x lim x 2 1 2 1 x 1 x 1 Since the left and right hand limits of fat x 1 Solution The function fis defined at all points of the real line Case 1 If c 1 then f c c 2 Therefore lim f x lim x 2 c 2 x c Thus fis continuous at all real numbers less than 1 Case 2 If c 1 then f c c 2 Therefore 3 1 3 3 2 1 2 2 1 1 1 3 3 Y X C 1 O X 1 2 3 Y Fig 5 3 ERT publiced 2 1 2 Example 11 Find all the points of discontinuity of the function f defined by x 2 if x 1 0 if x 1 x 2 if x 1 Solution As in the previous example we find that f is continuous at all real numbers x 1 The left hand limit of fat x 1 is 1 3 Since the left and right hand limits of fat x 1 do not coincide f is not continuous at x 1 Hence Fig 5 4 x 1 is the only point of discontinuity off The graph of the function is given in Fig 5 4 2 1 1 1 O 3 1 3 X 1 O 1 1 X F X 2 3

Use a efinite integral to find the area under the curve between the given x values f x 3x 6x 3 from x 1 to x 2 square units

Find the first and second derivative of the function G r T T G r G r

Differentiate the function B y cy 6 B y

1 Define 3 4 x if 2 x 2 if x 2 f x a Graph f from x 2 to x 8 1 f f x dx using the geometric interpretation of the integral and commonly 2 x 5 b Compute known area formulas Your work should make clear what regions you consider when computing your answer indicate these regions in your diagram for a

2 cos ma 1 1 1 z L Hopital Rule or Taylor series expansion 3 Evaluate lim 140 in terms of m where me R 0 You are not allowed to use 10 montiol

he velocity function is u t t2 3t 2 for a particle moving along a line Find the displacement and the distance traveled by the article during the time interval 3 6 isplacement istance traveled needed see page 287 of the Stewart Essential Calculus textbook for the definitions of these terms