Differentiation Questions and Answers

ASK YOUR TEACHER Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = 16x + 31, [-1,3] Yes, the Mean Value Theorem can be applied. No, because fis not continuous on the closed interval [a, b]. No, because fis not differentiable in the open interval (a, b). None of the above. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) - answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.) CH PRACTICE ANO f(b)-f(a) b-a (Enter your

The amount of garbage, G, produced by a city with population p is given by G = f(p). G is measured in tons per week, and p is measured in thousands of people. a. The town of Tola has a population of 30000 and produces 12 tons of garbage each week. Which equation gives this information in terms of the function f? Of(12)= 30000 Of (12) Of (30) = 12 Of (30000) = 12 1 30 b. What does the statement f (5) = 7 mean? A population of 5 people produces 7 tons of garbage each week. A population of 5000 people produces 5 tons of garbage each week. A population of 5000 people produces 7 tons of garbage each week.

Find all values of 0 (in rad) that satisfy the equation over the interval [0, 2π]. sin(---) sin(0) = sin smaller value larger value 8 = 0= = rad rad

Find each limit, if it exists. 2 - 9x X-∞ 2x³-8 (a) lim (b) (c) 2 - 9x lim X- 2x - 8 = 11 2 - 9x² lim x18 2x - 8 11

Consider the following function. f(x) = x² - 4x (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) X= (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) increasing decreasing (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum relative minimum (x, y) = =

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Identify the open intervals on which the function is increasing or decreasing. g(x) = x² - 6x - 280 Step 1 To determine the intervals on which the function is increasing or decreasing, first find the critical numbers of the given function. Determine g'(x). g(x) = x² - 6x - 280 g'(x)=2x-6 Step 2 To determine the critical numbers of g(x), set g'(x) equal to zero and solve for x. g'(x) = 0 2(x- 2x-6=0 1)=0 2r-6 X =

Use Newton's method with initial approximation x₁ = -1 to find x₂, the second approximation to the root of the equation x3 + x + 8 = 0. (Round your answer to four decimal place X₂ =

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.) " arctan(vx) new window √x

Simplify. sin²x sec²x - sec²x

Consider the following relation. {(-2, 2), (0, 5), (1, 6), (1, 7), (2, -1), (3, 1)) A. Is it a function? B. Is it a one-to-one function?

What is the amplitude of y = cos x? [?]

Let f be the function given by f(x)=x²-3x². What are all values of c that satisfy the conclusion of the Mean Value Theorem of differential calculus on the closed interval [0,3]? OA) 0 only OB) 2 only OC) 3 only OD) 0 and 3 OE) 2 and 3

Let f be a function defined and continuous on the closed interval [a,b]. Iff has a relative maximum at c and a<c<b, which of the following statements must be true? I. f'(c) exists. II. If f'(c) exists, then f'(c) = 0. IH. If "(c) exists, then f"(c) ≤0. OA) II only B) III only OC) I and II only OD) I and III only OE) II and III only

Convex quadrilateral ABCD has AB = 6, BC = 7, CD = 9, and DA = 5. The area of quadrilatearl of ABCD is 30. Determine the exact value of cos(ZDAB-ZBCD).

Perform the following operations on these complex numbers. (2-1)(1+3i)

Use a system of linear equations with two variables and two equations to solve. A moving company charges a flat rate of $80, and an additional $5 for each box. A taxi service would charge $17 for each box. Write an equation to represent the cost, y dollars, of moving a boxes with the taxi service. Ret

Let f (x) = 4x² - x f(x+h)-f(x) h Evaluate O 8x +4h-1 O 8x +4h+1 O 8x +4h-1-2 O 8x +4h+1 O4h-1 2x 04h-1-2/2 2x and simplify.

Let g(v) = v^14v Use logarithmic differentiation to determine the derivative of g. d/dv[g(v)]= Determine the slope of g at v = 1. g'(1) =

Let f(x)= x²(x − 4)^8/(x² + 2)^6 Use logarithmic differentiation to determine the derivative. f'(x) =

Use implicit differentiation to determine dy dx dy dx given the equation a² + y² = − 3. P

A rocket is launched vertically upward with an initial velocity of 29 ft/sec. Its height h in ft above the ground after t seconds can be represented by the following equation. How many seconds will it take the rocket to hit the ground? Round to the nearest hundredth. h = - 16t² +29t+2 coconds

(v + 3) (v + 7) (v-7)(v-2) Use logarithmic differentiation to determine the derivative of f. Let f(v) = dv [f(v)] =

Let g(y) = (y + 1)² (y + 3) (y — 9)³ Use logarithmic differentiation to determine the derivative of g. Dy[g(y)]=

9 u(u + 3) (u + 2) Use logarithmic differentiation to determine the derivative of f. d du Let f(u) = -[f(u)] =

Convert the following equation in rectangular form to polar form. A. r sin 0 = 2 B. r cos 0 = 2 C. r = 2 x = 2 D. cos 0 = 2

The function f(x) = is increasing on the open interval ( 4x³ + 15x² + 435.12z - 6.05 It is decreasing on the open interval (-∞, The function has a local maximum at ) and the open interval ( ,00).

Let h(w) = wln (15w) Use logarithmic differentiation to determine the derivative of h. Dw[h(w)]= =

Consider the function f(z)-5-47², -5 ≤ ≤2 The absolute maximum value is and this occurs at z = The absolute minimum value is and this occurs at x =

Solve the rational equation and identify the solution set. 6 ²-x-6 1 2-3-242 O a) {30} 20 Ob) {-2, 3} 000 d) {6}

Given the equation below, find dy dx = y = dy dx 24x7 + 9x³6y + y³ = 34 3 Now, find the equation of the tangent line to the curve at (1, 1). Write your answer in mx + b format

Find the critical numbers of the function f(x) = using a graph. I= I= I= is a Select an answer V is a Select an answer V is a Select an answer Question Help: Video Message instructor 125 75x+ 202³ + 2 and classify them -

Find the solution set of the equation. Use / to enter fractions, as in 3/4 means $³/1. Be sure to give a set as the answer. Do not enter any spaces in the answer. k+4 k+2 16 The solution set is N

Let h(x) = (17x) Use logarithmic differentiation to determine the derivative of h. d dx - [h(x)] = = Determine the slope of h at x = 1. h' (1) = =

Let f(x) = ln (1¹) f'(x) = ƒ' (e¹) =

Let f(x) = xcot (22x) Use logarithmic differentiation to determine the derivative of f. d dx -[f(x)] =

Let g(u) = 12"-2 Use logarithmic differentiation to determine the derivative of g. Dulg(u)] = Determine where g is differentiable. g is differntiable on

Let g(w) = 2ww-4 Use logarithmic differentiation to determine the derivative of g. d dw -[g(w)] = Determine where g is differentiable. g is differntiable on

w (w - 7)8 (w8 - 4)8 Use logarithmic differentiation to determine the derivative of h. Let h(w) = Dw[h(w)] =

Let h(v) = 23v Use logarithmic differentiation to determine the derivative of h. d - [h(v)] dv = Determine the slope of h at v = 1. h' (1) = =

Let F(x) = f(1³) and G(x) = (f(x))³. You also know that a = 9, f(a) = 3, f'(a) = 6, f'(a³) = 10 Then F'(a) and G'(a) = Question Help: Video Message instructor =

Let F(x) = f(f(z)) and G(z) = (F(z))². You also know that f(3) = 12, f(12) = 2, f'(12) = 13, f'(3) = 2 Find F'(3) - and G'(3) =

11v (v6 - 5) √v + 10 Use logarithmic differentiation to determine the derivative of f. Let f(v) = Dv[f(v)] =

19. Determine which line is perpendicular to the line 2x-3y + 17 = 0. a. 7 = (1,2) + s(3,2), s e R = $(3,2), SER = (2,-3) +-8(3,-2), s & R b. 7 = (1,7) + s(2,-3), s € R C. d.

2 √z(z + 2) Use logarithmic differentiation to determine the derivative of g. Let g(z) = = Dz[g(z)] =

Let h(z) = (z - 6)z-12+2 Use logarithmic differentiation to determine the derivative of h. Dz[h(z)] =

Let f(w) = (w + 8) +9 Use logarithmic differentiation to determine the derivative of f. d dw -[f(w)] = Determine where f is differentiable. f is differntiable on

Find the 38th derivative of the function f(x) = cos(x).

A six-digit number starts with a number between 6-9 in the first position, with no restrictions on the remaining 5 digits. a) Find the probability that a randomly-chosen phone number contains all different digits. b) Find the probability that a randomly-chosen phone number contains at least one repeated digit. a) Write an expression that models the probability. Select the correct choice below and fill in the answer box(es) within your choic (Type whole numbers.) O A. O C. •105 P5 1 • 105 The probability that a randomly-chosen number contains all different digits is. (Round to four decimal places as needed.) CELLE b) The probability that a randomly-chosen number contains at least one repeated digit is (Round to four decimal places as needed.) OB. O D. . P5 105 1 P5

Use implicit differentiation to find given the equation sin(xy) = y². dy dx dy dx =

The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s=- 8 where t is measured in seconds. Find the velocity (in m/s) of the particle at times t = a, t= 1, t=2, and t = 3. 2' t=a t=1 t=2 t = 3 V = V = V = V= m/s m/s m/s m/s