Limits & Continuity Questions and Answers

The function f(x) =x³ +8/x² has derivatives f(x) =x³ -16/x³ and f"(x)=48/x⁴ (a) Determine the x- and y-intercepts. (b) Determine the equations of any asymptotes and classify the behaviour at their endpoints. (c) Determine the intervals of increase and decrease and classify any local extrema. (d) Determine the intervals of concavity and identify any points of inflection. (e) Complete the 2nd derivative test. (f) Use this information to sketch the curve. Make sure all key features are clearly presented.

Determine the intervals on which k is continuous. Enter the solution using interval notation. k(x) = 3 cos x – 13 sin x k is continuous on ______

Sketch the graph of x²/6² - y²/4² = 1 selecting a type of graph, then select the center, then moving the cursor until the graph is correct).

A seagull, flying above a lake, dives for a fish according to the function h(t)=0.25t^3 -4t^2 +15t, where h(t) represents the height above the water, in feet, at t seconds. (A) Determine the velocity of the seagull when it exits the water. (B) Determine the acceleration of the seagull at 9 seconds.

Complete the table shown to the right for the half-life of a certain radioactive substance. Half-Life Decay Rate, k 18. 3 days

Determine the intervals on which g is continuous. Enter the solution using interval notation. g(x) =x +4/(x^2 - 4x - 32) g is continuous on ______________

The function f(x) = 7x +5 is one-to-one. a. Find an equation for f⁻¹,the inverse function b. Verity that your equation is correct by showing that f(f⁻¹(x)) =x and f⁻¹(f(x))=x.(Simplify your answer. Use integers or fractions for any numbers in the expression)

Determine the domain of the function of two variables f(x, y)= √(y+9x). The domain is {(x, y) | ___}. (Type an inequality. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the inequality.)

Please Simplify as instructed. 1 + sec(t)/1 + cos(t) Times top and button by 1 - cos(t) Simplify using results from #5 and #6 Simplify further using the result from #7 Simplify further so the answer has no fraction

R is the region bounded by the functions f(x) = 14 / (-1 + x) and g(x) = 2x – 14. Find the area of the region bounded by the functions on the interval [6, 8]. Enter an exact answer. Provide your answer below: A = ________units^2.

Let f(x) = x³+ 3x² – 1. Calculate the derivative f'(x) =________________ Calculate the second derivative f''(x) =________________ Note intervals are entered in the format (-00,5)U(7,oo) (these are two infinite intervals). On what interval(s) is f increasing? Increasing:_________________ On what interval(s) is f decreasing? Decreasing:____________________ On what interval(s) is f concave downward? Concave Down:____________________ On what interval(s) is f concave upward? Concave Up:____________ What is the limit as x approaches infinity? Limit is:______________ What is the limit as x approaches negative infinity? Limit is:-__________________- Sketch the graph and submit with your work.

Evaluate each limit: a) lim (x + x - 30) / (x - 6x + 5) x-3 b) lim {3 - √(1-4x)} / (x + 2) x->-2 c) lim (x +3) / ∛(x - 5) + 2 x->-3

Let f(x) = { (x² - 121) /(x - 11) if x ≠ 11 { a if x = 11 Determine the value of a such that f is continuous for all real numbers. a = ______

1) Use the graph off below to evaluate each limit, if it exists. If it does not exist, explain why. a) lim f (x) x->0 (b) lim f(x) x->1+ (c) lim f(x) x->1 2) Evaluate the limit, using limit laws. If it does not exist, explain why. (a) lim (5x³ - 3x² + 4) x->2 (b) lim x-9 / √(x-3) x->9 (c) lim (x⁴ - 1) /(x² - 1) x->1

Use the Limit Comparison Test to prove convergence or divergence of the infinite series. Σ∞ n=2 n²/n⁴ - 1 Σ∞ n=2 1/n(n-1) Σ∞ n=2 n/√n³+1 Σ∞ n=2 n³/√n⁷+2n²+1

Convert the angle to degrees, minutes, and seconds. 184.9258°.

Question 31 Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 29°41'=__________

Let h(x) = { eˣ+¹ if x > -1 { ax + 2 if x≤ -1 Determine the value of a such that his continuous for all real numbers. a = ____

Determine the intervals on which r is continuous. Enter the solution using interval notation. r(x) =12x/ (400 – x^2)^2 r is continuous on ______________

A bag has seven balls labeled A, B, C, D, E, F, and G. One ball will be randomly picked, and its letter will be recorded as the outcome. Give the sample space describing all possible outcomes. Then give all of the outcomes for the event of choosing a vowel. If there is more than one element in the set, separate them with commas.

Determine and classify the discontinuities, if any, of g. g(x) = 4cosx + 5x . State the removable discontinuities, if any, of g. If multiple removable discontinuities exist, enter the solutions using a comma-separated list. (a) g has a removable discontinuity at x = __ g has no removable discontinuities. State the non-removable discontinuities, if any, of g. If multiple non removable discontinuities exist, enter the solutions using a comma-separated list. (a) g has a non-removable discontinuity at x = __ g has no non removable discontinuities.

Let h(x) = { 5x² if x > 5 { -x + a if ≤ 5 Determine the value of a such that h is continuous for all real numbers. a = ____

When a skydiver free-falls from an airplane, her downward velocity in feet per second is given as v(t) = 240(1 – 0.85'), where t is the number of seconds since the jump. a. First, find the limit of v(t) as t -> ∞. This is the terminal velocity of the jumper.

Use the Direct Comparison Test to determine whether the infinite series is convergent. Σ∞ n=1 1/n2ⁿ Σ∞ n=1 n³/n⁵+4n+1 Σ∞ n=1 1/n¹/³+2ⁿ Σ∞ n=1 1/√n³+2n-1 Σ∞ n=1 4/m!+4ᵐ Σ∞ n=1 √n/n-3

Using the graph below, determine the following: (a) lim f(x) (c) lim f(x) (b) lim f(x) (e) lim (x) (d) lim f(x) (f) lim f(x)

In Exercises, determine whether the series converges absolutely, conditionally, or not at all. Σ∞ n=1 (-1)ⁿ‾¹/n¹/³ Σ∞ n=1 (-1)ⁿn⁴/n³+1

If a= 3, a₂=0 and an=3an-1 + 3an-2 then find the value of a₆.

Does the table represent a linear or exponential function? Find a formula x 1 2 3 4 h(x) 70 49 34.3 24.01

Use the derivative formula f'(x)= lim h->0 f(x + h) – f(x)/h to find the derivative of f(x) = √x Find the derivative of y = 4x² - 3x/sinx

Use the function above to find the following values of f(x). Show all of your work for full credit. f(x) = {5/3 x + 3 for x ≤ -3, {-x -5 for -3 < x <, {5x - 19 for x ≥ 2 a. f(-3) b. f(1) c. f(2) d. f(4)

Consider the function f(c) = 7(x - 5)^(2/3). For this function there are two important intervals: (- ∞, A) and (A, ∞) where A is a critical number. A is _______________ For each of the following intervals, tell whether f(x) is increasing or decreasing. (- ∞, A): ____________ (A, ∞): _____________ For each of the following intervals, tell whether f(x) is concave up or concave down. (-∞, A): ____________ (A, ∞):_____________

Suppose sin 2x =-√ 2/2 Find all solutions 0≤x≤2π . Give exact values in radians,and type "pi" for π. x=____________ (If there is more than one answer, enter the answers separated by commas)

The following formula is used by psychologists and educators to predict the reading ease, E, of a passage of words where w is the number of syllables in a 100-word section and s is the average number of words per sentence. Find the reading ease where w= 141 and s=5. E = 206.835 -0.846W - 1.015s

What is a coterminal angle of 22π/3? (a) 60° (b)120° (c) 240° (d)300°

Solve sin^2(x) = 4 cos(x) for all solutions 0 ≤ x < 2π. x = Give your answers as values accurate to at least two decimal places in a list separated by commas.

Let F be the set of functions of the form f(x) = A sin(x) + B cos(2x), where A, B are some real constants. Show that there must exist exactly one function f in F so that for any f∈F √∫(f(x)- arctan(x))²dx≤ √∫(f(x)- arctan(x))²dx

Triangle EFG has vertices El-11), F(3.1) and G(45) Find the coordinates of the image of point Fafter a reflection across the x-axis.

The function f(x)=- 2x^3 +27x^2 - 108x + 10 has one local minimum and one local maximum. This function has a local minimum at x =_______________ with value ___________ and a local maximum at x =___________ with value______________

Jocelyn invested $56,000 in an account paying an interest rate of 7(3/8)% compounded continuously. Oliver invested $56,000 in an account paying an interest rate of 7(1/4)% compounded monthly. After 13 years, how much more money would Jocelyn have in her account than Oliver, to the nearest dollar?

The function f(x) = 2x^3 – 27x² + 84x – 3 has two critical numbers. The smaller one is x = and the larger one is x =

Determine the Cartesian equation of the plane that contains the points A(-2, 3, 1), B(3,4,5), and C(1, 1, 0).

Harold uses the binomial theorem to expand the binomial. (a) What is the sum in summation notation that he uses to express the expansion? Write the simplified terms of the expansion.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = In(x), y = 0, x = 5; about the y-axis dx

Find the average value ha of the function h on the given interval. ave have = h(u): 12 In(u) U (3.78656) [1, 7]

An acute angle, o, is in a right triangle such that cosθ=24/25. What is the value of csc θ? (a)7/24 (b)7/25 (c)25/7 (d)25/24

lim (cos²θ - sin²θ)/(cosθ - sinθ) = θ->π/4 (A) - 1 (B) 0 (C) √2 (D) 1 (E)1/√2

Use the table below to find the limit, if it exists. If the limit does not exist type DNE. If you get infinity, you must type out the entire word, for negative infinity type-infinity. Bolded terms must be typed as you see them here or Schoology will count it wrong. Use the table below of the rational function y = H(x) to find the indicated limits. For limits that do not exist, write D.N.E. x H(x) -1000 -0.998 -4.001 0.666 -4 undefined -3.999 0.666 1.999 -4497 2 undefined 2.001 4504 1000 3.0002 A. limH(x)= x->-4(from left) B.limH(x)= x->-4(from right) C.limH(x)= x->-∞ D.limH(x)= x->-4 E.limH(x)= x->2(from left) F.limH(x)= x->2+

Find the antiderivative for the function f(x) = 3x(x² + 1)⁴dx.

If a curve is defined by the parametric equations x(t) = 2t and y(t) = -4t² - 2 over the interval - 2 ≤ t ≤ 2, which of the following gives the correct equation and domain of the path of the particle? (A) y = -2x² - 1, -2 ≤ x ≤ 2 (B) y = -x² - 2, - 2 ≤ x ≤ 2 (C) y = -x² - 2, -4 ≤ x ≤ 4 (D) y = x² - 2, -2 ≤ x ≤ 2 (E) y = x² - 2, -4 ≤ x ≤ 4

The graph of the equation x² + y² - 2x - 6y = 21 is: (A) a circle (B) an ellipse (C) a hyperbola (D) a parabola (E) a sphere