Limits & Continuity Questions and Answers

If the odds against E occurring are 3:5, then find P(E) and P(E'). P(E)= (Simplify your answer.) P(E')= (Simplify your answer.)

Suppose H (x)=7x²-3. Find two functions fand g such that (fog)(x) = H (x). Neither function can be the identity function. (There may be more than one correct answer.)

Solve for x, using the inverse trigonometric functions. 2 cos x = 1 O x = 1/3 O x = 1/₂ O x = 74 O x = n/6

Suppose you deposit $1000 at 4% interest compounded continously. Find the average value of your account during the first 3 years.

Evaluate the limit: lim h→0 (−5+h)²-25 h

Which grid is used to graph polar equations? O Coordinate grid Rectangular system O Polar grid All of the choices

For the real-valued functions f(x) = x +4 and g(x) = 4x-7, find the composition fog and specify its domain using interval notation. x + 3

f(x) = 5x-4 g(x) = √2x+3 Find f g and f-g. Then, give their domains using interval notation.

12.) (a) lim sin x X → 0 (b) lim 1 x² sin x cos x cos x tan x x → 0 tan x

At one point, license plates in a certain state consisted of 5 letters (excluding I and O), followed by 2 numbers. Complete parts (a) through (e) below. a) How many plates are possible when letters and numbers can repeat? Set up the expression that can be used to calculate the number of plates that are possible when letters and numbers can repeat The expression is 244.10² (Do not simplify) The number of plates possible when letters and numbers can repeat is (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.) b) How many plates would be possible if letters could repeat but numbers could not? (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.) c) How many plates would be possible if letters could not repeat but numbers could? (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.) d) How many plates would be possible if neither letters nor numbers could repeat? CELE (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to three decimal places as needed.) e) What is the probability that a randomly generated plate has no repeating letters or numbers? (Type an integer or decimal rounded to three decimal places as needed.)

An ordered pair of a function is given as (x, y). coordinates to find the corresponding pair of its inverse. Swap O Invert Subtract Add the x and y

Q1. The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. Find the (a) linearization (Taylor polynomial of order 1) and (b) the quadratic approximation of the following function f(x) at x = π. (c) Find lim f(x) using (1) L'Hopital's Rule and (2) the linear approximation you found in (a). Discuss your x-0 findings. (15 points) f(x) = sin x X

4. y=-3+2 sin(x+²) amplitude: a) b) period. c) graphing interval that will include one period: d) Range. -Basic function: -including vertical stretch/shrink and shift. e) graph all transformations starting with y=sin(x + = = = ), then y = -3+2 sin(x + 11 )

If f(x) = x + 10/3, find f-¹(x). Of(x) = x + 10/3 Of(x) = x - 10 O f-¹(x) = x - 10/3 f(x) = x + 10

The distance between [(3 - 2i) * (8 - 1)] and (9 + 2i) is O √620 O √610 O √54 √640

Suppose that f(x) is a continuous function with f(2)= - 10 and f(6) = 10. Determine which choice best describes the following statement. "f(x) = 0 for some x in the interval [2, 6]" Always true O Always false O Sometimes true and sometimes false

Which of the following is a vector equation of the equation below: 2+2 = 3-1 -7 Ox= -2-7t y = 1 + 3t Ox=2+7t y = -1 -3t O[x,y] [2,-1] + [7.-3] O[x,y] = [-2,1] + [-7,3]

Estimate the limit numerically or state that the limit does not exist (DNE): sin (9x) lim x-0

Find the limit of the transcendental function. 9.) (a) lim sin 2x 5π 6 (b) lim cot X→ 2 TIX

fv-x+5 g(x) = √5x-3 f Find f-g and Then, give their domains using interval notation. 9

For the real-valued functions f(x) = 4x + 1, and g(x)=√x-2, find the composition f g and specify its domain using interval notation.

Suppose H (x)=(9-6x)4. Find two functions f and g such that (fog)(x) = H (x). Neither function can be the identity function. (There may be more than one correct answer.) f(x) = 0 g(x) = 0 8 0/0 X

Given the equation y=8 sin(5(x-4))+2` The amplitude is: 8 The period is: 2 The horizontal shift is: 4 The midline is: y = 2 X OT units to the Right

The point A(3, 9) lies on the curve y=x² If Bis the point (x, x^2), find the slope of the secant line AB for the following values of x. Use all decimal places in your response. If x= 3.1, the slope of AB is: A/ If x= 3.01, the slope of AB is: A/ If x= 2.9, the slope of AB is: A/ If x= 2.99, the slope of AB is: N Based on the above results, guess the slope of the tangent line to the curve at A(3, 9): (Integer value expected.)

4.) Evaluate the given piecewise - defined function for the indicated values. f(x) = x² + 1, -x² + 4x, -X (a) Find: f(3) if x ≤ 1 if x > 1 (b) Find: f²) (c) Find: f(3x + 2); X < 3

Solve using the quadratic formula. 03x²9x - 5 X = List your answers, separated by commas. Give exact answers or decimals accurate to at least 3 decimal places. O 10 386832 s Popesc

In Exercises 17-22, find the sum of the first terms of the sequence. The sequence is either arithmetic or geometric. 17. 2.5.8....: n=10 18. 14. 8. 2.:=9 1 2 3 3 21 ...:^= 11 19.4.-2.1. 20.6.-3. 21. -1, 11, 22.-2, 24, ...:=12 4' 121.... 9 288....=8

Find the remainder when x³ + 6x² + 4x - 3 is divided by x - 2 using remainder theorem. 37 0 -37 45

Which of the following is a scalar? a parachutist falling at 20 km/h downward O a car travelling at 50 km/h to the east a man's mass of 88 kg Jhry a child pulling a wagon with a force of 1 N at 30° to the horizontal

.In a field PQRS, which is in the shape of a trapezium, PQ|| RS, PQ-58 m and RS-24 m. A triangular flowerbed XQR is cut in such a way that, the shape of the remaining field becomes a parallelogram PXRS. If the area of the field PQRS is 779 m², find the area of the flowerbed and the area of the remaining field.

Solve sin² (t) t= - 7 cos(t) for all solutions 0 ≤ t < 2π Give your answers accurate to 2 decimal places, as a list separated by commas

1- Given a geometric sequence whose first term is 15 and common ratio is - 2, find the third term. -120 -60 60 11

Use the limit Theorem to find the following limits, if they exists. 5.) (a) lim (2x³ - 3x² + 5x) X→ 2 (b) lim X→ 0 2x² X 3x

Solve the equation: m² + 12m = 0 Answer: m = Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get 4 and 2 as your answers, then enter 4,-2/3 in the box. 3

) (a) lim X→ 2 +² - 7x + 10 X - 2 (b) lim X → -5 x² + 8x + 15 X + 3

Let y = 2(x - 5)2 - 6. Part A: Is the given relation a function? Is it one-to-one? Explain completely. If it is not one-to-one, determine a possible restriction on the domain such that the relation is one-to-one. (5 points) Part B: Determine y¹. Show all necessary calculations. (5 points) Part C: Prove algebraically that y and y¹ are inverse functions. (5 points)

The conjugate of 2 + 3i is O3+2i 2/31 2-3i O-2-3i

Find x in the following equation. X= logbx + log(x-6) = log27

Solve 12 cos² (w) - 10 cos(w) + 2 = 0 for all solutions 0 <w< 2π W= Give your answers accurate to 2 decimal places, as a list separated by commas

A polynomial or integer that is left over after one polynomial or integer is divided by another polynomial or integer is called dividend divisor quotient remainder

A website requires users to create a password that is 7 characters long. Each character can contain numbers, or symbols chosen from ? , #, and 1. The password does not have to include both numbers and symbols. a) How many 7-character passwords are possible if numbers and symbols can be used more than once? b) How many 7-character passwords are possible if no numbers or symbols can be repeated? c) What is the probability that a randomly generated 7-character password contains 7 different characters? C a) Set up the expression that can be used to calculate the number of possible 7-character passwords if numbers and symbols can be used more than once. The expression is (Do not simplify.) The number of different passwords that are possible if numbers and symbols can be used more than once is (Simplify your answer.) b) The number of different passwords that are possible if no numbers or symbols can be repeated is. (Simplify your answer.) A c) The probability that a 7-character password contains 7 different characters is (Type an integer or decimal rounded to three decimal places as needed.)

The set of points (2et, t), where t is a real number, is the graph of y = Hint: Think inverse. 1 2xe x 01/1/20 O e²x Onx-m² In x - In 2 O 2e 1 X In x - In 2 it

Solve 6 cos(4x) = 3 for the smallest three positive solutions. Give your answers accurate to at least two decimal places, as a list separated by commas

Find x in the following equation. log 10(x+1) - log 10 (x-1)= 1 X= (Type a fraction or an integer. Simplify your answer.)

Solve 7 sin T = (5₁) = 2 for the four smallest positive solutions Give your answers accurate to at least two decimal places, as a list separated by commas

Glenn and Kim are members of a comedy troupe that has 9 members in all. How many ways can the 9 members line up if Glenn and Kim must stand side by side? CITE Give the expression that can be used to calculate the number of ways the members of the troupe can line up. Choose the correct answer below. OA. 91 OB. 71-8 OD. 71 OC. 91.8 OE. 91.2 OF. 91.8.2 O G. 71.2 OH. 71.8.2 The number of ways that the troupe can line up is (Simplify your answer.) M

Solve the equation 12y² - 17y + 6 = 0. Answer: y = Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get 4 and - as your answers, then enter 4,-2/3 in the box. 2 3

The engines of a plane are pushing it due north at a rate of 300 mph, and the wind is pushing the plane 20⁰ west of north at a rate of 40 mph. What is the magnitude of the resultant vector? [?] mph Round to the nearest tenth.

Let f(t)= (1+t, 3, t, 7)||. What is the minimum value of this function of ? Answer the same question for the function g defined by g(t)= ||(1,3,7-t)ll. (10%)

Let f(x) = ²x Determine the equation of the tangent line to f at x = 1 2. Report the solution using slope-intercept form.