Application of derivatives Questions and Answers

Determine the equation of the tangent to y = (3x-2 - 2x³)5 at (1, 1).

Let s(t) = 6t³ - 27t² - 180t be the equation of motion for a particle. Find a function for the velocity. v(t) = - Where does the velocity equal zero? [Hint: factor out the GCF.] t and t = Find a function for the acceleration of the particle. a(t) =

Simplify sin² (t) to an expression involving a single trig function with no fractions. 1 – sin²(t) If needed, enter squared trigonometric expressions using the following notation. Example: Enter sin² (t) as (sin(t))².

Find x in the following equation. logbx+ logb(x-2) = log24 X=

What is - 2 x (8 - i)? - 16 + 2i O-16-2i 16 + 2i - 10 + 2i

Solve cos(x) There are two solutions, A and B, with A <B A = -0.25 on 0x< 2π B = Give your answers accurate to 3 decimal places

A spherical balloon is inflated with gas at a rate of 700 cubic centimeters per minute. (a) Find the rates of change of the radius when r = 70 centimeters and r = 85 centimeters. r = 70 r = 85 1 88 dr dt x cm/min cm/min (b) Explain why the rate of change of the radius of the sphere is not constant even though dv/dt is constant. The rate of change of the radius is a cubic relationship. dv The rate of change of the radius is a linear relationship whose slope is dt depends on r², not simply r. dr as a function runs parallel to the volume function, which is not linear. dt The volume only appears constant; it is actually a rational relationship. Xx

Determine the points on the curve y = x³ x² - x + 1 where the tangent - is horizontal.

Consider the following function. f(x) = (x + 6)2/3 (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 you 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 relative maximum (x, y) = relative minimum (x, y) =

Determine a. y = dy dx for each of the following functions: c. y = ε+ ε - b. y = 6(2x - 9)5 2 √x 3x5 + 4π + X V3 + 6√3/x I

Find the point where the tangent line to y = x³ at x = 0.3 crosses the x-axis. x-intercept =

If f(x) = x³ + 3x² + 4x + 5 and g(x) = 5, then g(f(x)) = O5x³ + 15x² + 20x + 25 225 5x² + 15x + 25 05 1125 A

Let h(x) = e-4 Determine the equation of the tangent line to h at x = -1. Report the solution using slope-intercept form. Submit Question

Find the slope of the tangent line to the curve -3x² + xy + 2y³ = - 24 at the point (4, 2). Slope

Use the given information to answer the following questions. center (3, -2, 5), radius (a) Find an equation of the sphere with the given center and radius. (b) What is the intersection of this sphere with the xz-plane? , y = 0

Solve sec(4a)-8= 0 for the four smallest positive solutions B

Solve 7 sin (1²) = 2 for the four smallest positive solutions

The radius r of a circle is increasing at a rate of 4 centimeters per minute. Find the rate of change of the area when r = 29 centimeters.

Determine an angle that is a coterminal angle of the angle 0. Use degree measure. (Answers may vary.) -1.0 -0.5 y 1.0 0.5 -0.5 1.0 0= 117 12 0.5 1/0 X

A car travels 13 km southeast and then 16 km in a direction 40° north of east. Find the direction of the car's resultant vector. [?]°

Consider - 10 In (y) = 6y sin æ Determine the equation of the tangent line at (0, 1). Report the solution using slope-intercept form.

Consider Determine the derivative of y with respect to x. = 5x + y x - 2y = x7

The radius r of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rate of change of the volume when r = 12 inches. in.³/min (b) Find the rate of change of the volume when r = 38 inches. in.3/min

Let f(x) = x²√3+x² Determine the equation of the tangent line to f at (1, 2). Report the solution using slope-intercept form.

1 (5 - 8x) ³ Determine the derivative of h. Let h(x) = h'(x) = Determine the slope of h at x = -6. h'(- 6) =

Find the equation of the line tangent to the graph of x³ + y³ = 5xy — 11 at (1, − 3). The equation of the tangent line is

Problem 8. A baseball is launched into the air with an upward velocity of 106 feet per second. The height of the baseball after t seconds can be represented by the function f(t) given by f(t) = -16t² + 106t + 9.3. (Round all your answers to three decimal places, if needed.) a) Find the time it takes for the baseball to reach its maximum height. b) What is the maximum height? c) When will the baseball hit the ground?

D Shrives Publishing recently reported $13,000 of sales, $5,500 of operating costs other than depreciation, and $1,250 of depreciation. The company had $3,500 of bonds that carry a 6.25% interest rate, and its federal-plus-state income tax rate was 35%. During the year, the firm had expenditures on fixed assets and net operating working capital that totaled $1,550. These expenditures were necessary for it to sustain operations and generate future sales and cash flows. What was its free cash flow? (Round your intermediate and final answers to whole dollar amount.) a. $4,704 b. $2,898 c. $3,161 d. $3,086 e. $3,763 3

Your company has just taken out a 1-year installment loan for $72,500 at a nominal rate of 20.0% but with equal end-of-month payments. What percentage of the 2nd monthly payment will go toward the repayment of principal? O a. 71.70% Ob. 97.55% O c. 83.37% d. 101.72% e. 86.71% 3

You are considering investing in a bank account that pays a nominal annual rate of 7%, compounded monthly. If you invest $3,000 at the end of each month, how many months will it take for your account to grow to $310,000? a. 72.18 b. 81.11 c. 90.84 O d. 93.27 Oe. 95.70 1

Sean gets an average of 20 calls during his 8 hour work day. What is the probability that Sean will get at most 9 calls in a 4 hour portion of his work day? Round the final answer to three decimal places. Provide your answer below:

Find an equation to the tangent time for the graph of/ at the given point f(x)=(2x³ + 8), (1, 100)

After graduation, you plan to work for Dynamo Corporation for 12 years and then start your own business. You expect to save and deposit $7,500 a year for the first 6 years (t = 1 through t = 6) and $15,000 annually for the following 6 years (t = 7 through t = 12). The first deposit will be made a year from today. In addition, your grandfather just gave you a $32,500 graduation gift which you will deposit immediately (t = o). If the account earns 9% compounded annually, how much will you have when you start your business 12 years from now? a. $292,914 b. $286,936 c. $310,848 d. $269,003 e. $298,892

An office copier has an initial price of $2,300. A service contract costs $200 for the first year and increases $60 per year thereafter. It can be shown that the total cost of the copier after n years is given by C(n)=2,300+ 170n + 30n². C(n) The average cost per year for n years is given by C(n) = n (C) When is the average cost per year at a minimum, and what is the minimum average annual cost? [Hint: Refer to the sketch in part (B) and evaluate C(n) at appropriate integer values until a minimum value is found.] The average cost per year is at a minimum in approximately years. (Type a whole number.) The minimum average annual cost is approximately $ per year. (Round to the nearest cent as needed.) (...) g The average annual cost is minimum in approximately (Round to the nearest integer as needed.) (D) Graph the average cost function on a graphing calculator and use an appropriate command to find when the average annual cost is at a minimum. years. € O

Use the Principle of Mathematical Induction to show that the following statement is true for all natural numbers n. 1 9+8+7+...+(10-n) = n(19-n) What two conditions must the given statement satisfy to prove that it is true for all natural numbers? Select all that apply. The statement is true for the natural number 1. The statement is true for any two natural numbers k and k + 1. If the statement is true for the natural number 1, it is also true for the next natural number 2. If the statement is true for some natural number k, it is also true for the next natural number k +1. *** Show that the first of these conditions is satisfied by evaluating the left and right sides of the given statement for the first natural number. 1 9+8+7+...+(10-n) = 2n n(19-n) 9 = 9 (Simplify your answers.) To show that the second condition is satisfied, write the given statement for k+ 1. 9+8+7++ (10-k) + (Use integers or fractions for any numbers in the expression. Type your answer in factored form.)

Prezas Company's balance sheet showed total current assets of $2,500, all of which were required in operations. Its current liabilities consisted of $975 of accounts payable, $600 of 6% short-term notes payable to the bank, and $250 of accrued wages and taxes. What was its net operating working capital? a. $1,199 b. $1,173 c. $1,326 d. $1,275 e. $1,301 1

Problem 5. Find the standard form of the equation of the circle x² + y² - 2x +12y + 12 = 0. Then, state the center and radius of the circle.

Problem 1. If f(x) has an inverse function f-¹(x), could either the graph of f(x) or the graph of f¹(x) be symmetric with respect to the y-axis? Explain your reasoning or use an example to illustrate your answer.

Your Aunt Ruth has $540,000 invested at 6.5%, and she plans to retire. She wants to withdraw $40,000 at the beginning of each year, starting immediately. How many years will it take to exhaust her funds, i.e., run the account down to zero? O a. 24.55 b. 28.41 O c. 25.65 d. 30.62 e. 27.58 1

A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 25°. From a point 1000 feet closer to the mountain along the plain, they find that the angle of elevation is 29° How high (in feet) is the mountain? F

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 5.3 mi apart, to be 35 and 51°, as shown in the figure. A NOTE: The picture is NOT drawn to scale. 35° Find the distance of the plane from point A. distance from A = mi Find the elevation of the plane. height= mi 5.3 mi 51° B

Graph the following function over the indicated interval. y=2*; [-3,3] Choose the correct graph below. O A. Ay O B. -4 Ay 10- Q ⒸO Q C... O C. Ay Q KÜLA Q -4 O D. Ay O

Assume Za is opposite side a, ZB is opposite side b, and Zy is opposite side c. Solve triangle ABC if ZA = a = 36.9°, b = 12.6 mi, and c = 15.8 mi. Using the Law of Cosines, a≈ mi. Your answer should accurate to 2 decimal places. Using the Law of Cosines again, cos/B= cos B Your answer should accurate to 5 decimal places. Thus, B Your answer should accurate to 2 decimal places. cos/C= cos y Your answer should accurate to 5 decimal places. Thus, y Your answer should accurate to 2 decimal places

A water tower is located 412 feet from a building. From a window in the building, an observer notes that the angle of elevation to the top of the tower is 35 and that the angle of depression to the bottom of the tower is 26°. How tall is the tower? height = How high is the window from the ground? window is feet. feet above the ground. Report answer accurate to 2 decimal places. If it helps, there is a similar picture on page 486 of your Stewart text (see # 55).

5. Let C be the curve of intersection of the hyperbolic paraboloid z=x²-y² and the cylinder x² + y² = 1. (1) Find the curvature of C at the point P(1,0,1). (2) Find the osculating plane at the point P(1,0,1).

The function f(x) = (7x - 9)e (7x - 9)e-2 has one critical number. Find it.

we know barycentric coordinates is defined as : (ao +...+an)OP = ao OA++ an OAn, below is two results of the coordinates, please point out OP and OAO OA1 -- OAn Barycentric coordinates ... (1,0,0) (1,0,0) (1/2,1/2,0) (1/2,1/4,1/4) (1/4, 1/2,1/4) (1/2,0,1/2) (1/3,1/3,1/3) (1/4, 1/4, 1/2) (0,0,1) (1/2,1/2,0) (0,1/2,1/2) (1/2,1/4,1/4) (1/4,1/2,1/4) (1/3,1/3,1/3) (0,1,0) (O (0,1,0) (1/2,0,1/2) (1/4, 1/4, 1/2) (0,1/2,1/2)

Susan is six years older than Beth. In four years, Susan will be twice as old as Beth. How old is Susan now?

Using the information from the table below, draw the graph for a function f. Label all intercepts, horizontal and vertical asymptote(s), inflection point(s) for full credit. Domain of f I and y coordinates of r- intercepts of f r and y coordinates of y-intercepts of f Equation(s) of vertical asymptote(s) Equation(s) of horizontal asymptote(s) Critical number(s) Open intervals where f is increasing Open intervals where f is decreasing r and y coordinates of all local minimum(s) of f r and y coordinates of all local maximum(s) of f Open intervals where f is concave up Open intervals where f is concave down r and y coordinates of all inflection point(s) of f (-∞, 0) U (0,0) (-1,0), (,0) NONE x=0)) y=-2 -1, 1 (-∞, -1), (1,00) (-1,0), (0,1) (1,-4) (-1,0) (-∞, -√2), (0, 1) (-1,0), (1,0) (√2.-3.7) and (-√2,-0.2)

Consider the curve given by y5 + 1 = ln(x + 5). a. [2 pts] Use a computer to graph the curve. Make sure to include most interestin features in your graph. b. [4 pts] Set up and simplify an integral in terms of x that represents the arclength from x = -2 to x = 2. Then evaluate your integral using a computer system. c. [4 pts] Set up and simplify an integral in terms of y that represents the arclength from x = -2 to x = 2. Then evaluate your integral using a computer system.Make sure it agrees with part a.