Application of derivatives Questions and Answers

A highway and a railway cross at and angle of 22 degrees. A passenger train is approaching the intersection at 74 miles per hour while a car is approaching the intersection at 80 miles per hour. Determine the rate of change of the distance between the passenger train and the car when the passenger train and the car are 5 and 1 miles away from the intersection, respectively. Round the solution to the nearest ten-thousandth, if necessary. miles per hour car passenger train

A 5 foot tall man jogs at a rate of 23 feet per second toward a lamppost that is 19 feet tall. Determine the rate of change of the tip of the man's shadow when he is 21 feet away from the lamppost. feet per second Determine the rate of change of the length of the man's shadow when he is 21 feet away from the lamppost. feet per second

Solve for x where T ≤ x ≤ 2π sin²x - cos²x cos²x [?]T = 0 = π

Perform the following operation, and write the answer in scientific notation with the correct number of significant figures. 0.00851.3= [?] x 10

Solve for x by factoring where 0≤x≤T. tan²x - tan x = 0 0, [2] T 7 Enter the next smallest value.

Corn is falling off a conveyor belt onto a conical pile at the rate of 10 cubic feet per minute. The diameter of the base of the pile is 4 times the height. Determine the rate at which the height of the pile is changing when the height is 31 feet.

Let f(x) = - 3 cos x for 0 ≤ x ≤ π. Determine the derivative of f-¹ at x = (1-¹)-(-3√³) - ( 2 3√3 2

Perform the mathematical operation and round the answer to the appropriate number of significant figures. 24.32 x 0.010 = [?

The radius of a sphere is increasing at a rate of 2 centimeters per hour. Determine the rate of change of the sphere's surface area when the radius is 25 centimeters. square centimeters per hour

The radius of a circle is increasing at a rate of 10 inches per minute. Determine the rate of change of the circle's area when the radius is 41 inches. square inches per minute

A 6 foot tall woman walks at a rate of 6 feer per second toward a lamppost that is 15 feet tall. Determine the rate of change of the tip of the woman's shadow when she is 9 feet away from the lamppost. feet per second Determine the rate of change of the length of the woman's shadow when she is 9 feet away from the lamppost. feet per second

The radius of a sphere is increasing at a rate of 3 centimeters per hour. Determine the rate of change of the sphere's volume when the radius is 16 centimeters. cubic centimeters per hour

Let h(x) = - 2x³ - 3 Determine the derivative of h-¹ at x = – 5. (h-¹) '(-5) =

The point P(3, -3) lies on the curve y = (a) If Q is the point x, (i) 2.9 MpQ (ii) 2.99 MpQ = E (iii) 2.999 mpQ (iv) 2.9999 MPQ (v) 3.1 MPQ (vi) 3.01 MPQ = (vii), 3.001 MPQ = 3 (x₁__²_-). (viii) 3.0001 m = PQ 3 2-x find the slope of the secant line PQ (correct to six decimal places) for the following values of x.

Suppose x and y are both differentiable functions of t and x³ + y² = c5 where c is a constant. Determine dy dt = dy dt " given dx dt 23 3, x=-3, and y = 1.

Air traffic control detects two airplanes at the same altitude converging on a single point as they fly at right angles to each other. One plane is 152 miles away from the point, flying at 420 miles per hour. The other plane is 714 miles away from the point, traveling at 445 miles per hour. At what rate is the distance between the two planes changing? miles per hour

Let g(w) = arctan( - 6w) Determine the derivative of g. g'(w) = = Determine the slope of g at w = 9. g'(9) =

Suppose both f and f-1 are differentiable functions such that . • f(7) = 8 • f'(7) = 7 9 Determine the equation of the tangent line of f¹ at x = 8. Report the solution using slope-intercept form. -

Let g(x) = (x² - 8x + +3)³ Determine the equation of the tangent line to g at x = 7. Report the solution using slope-intercept form.

Let h(z) arccot ( 122³ +212²) - Determine the value(s) of z, if any, for which h has a horizontal tangent line. Oh has a horizontal tangent line at z = Oh has no horizontal tangent lines.

Let f(x) = Determine the derivative of f. f'(x) = Determine the slope of fat x = 3. f'(3) = = - 3 cot(8x)

For the given function, (a) find the slope of the tangent line to the graph at the given point; (b) find the equation of the tangent line. 9 h(x) = -atx=7 X (a) The slope of the tangent line at x = 7 is.

Suppose the line y = 5x - 9 is tangent to the curve y = f(x) when x = 3. If Newton's method is used to locate a root of the equation f(x) = 0 and the initial approximation is x₁ = 3, find the second approximation x2 x2 =

Assume that .x, y, z are positive. Use Lagrange multipliers to find the maximum of the function f(x.y.z)= e^xyz subject to the constraint x² + y² + = 9.

Consider the following function. f(x) = x2-49 (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 (x, y) = relative minimum (x, y) =

Find the rate of change of y with respect to x at the indicated value of x. y csc(x) - 6 cos(x); x=π/6 y'=

Consider the following function. x2 x² - 49 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) x = (0,0) = (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.) (-∞, -7) U(-7,0) (0,7) U (7,00) increasing decreasing X (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = (DNE * ) relative minimum (x, y) = DNE )

Consider the function on the interval (0, 2). f(x) = sin(x) cos(x) + 2 (a) Find the open interval(s) on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing (b) Apply the First Derivative Test to identify all relative extrema. relative maxima (x, y) = (smaller x-value) relative minima (x, y) = (x, y) = (x, y) = ( (larger x-value) (smaller x-value) (larger x-value)

Consider the function below. f(x) = -4x² + 24x + 2 (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 answer using interval notation. If an answer does not exist, enter DNE.) increasing decreasing (c) Apply the First Derivative Test to identify the relative extrema. (If an answer does not exist, enter DNE.) relative minimum -C relative maximum (x, y) = (x, y) =

Consider the following function. f(x) = 5x + 1 x (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (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 (x, y) = relative minimum (x, y) =

Consider the following function. f(x) = (5x)(x + 1)² (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 (x, y) = relative minimum (x, y) =

No, because p cannot be negative P = 84 Yes, because each value of a corresponds to exactly one value of p OYes, because the relation defines p in terms of s O No, because some values of p correspond to more than one value of s

Consider the following function. f(x) = x² - 32x + 2 (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 (x, y) = relative minimum (x, y) =

Consider the function on the interval (0, 2). f(x) = sin x + cos x (a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing (b) Apply the First Derivative Test to identify the relative extrema. relative maximum (x, y) = relative minimum. (x, y) =

Simplify. sin x sec x - sin x csc x [?]x - [ ]

If the rate of change of T is proportional to (30+ T), where T > -30. If T(0) = 4, and T(2) = 6, find T(5).

Consider the measurements shown. The most precise value is highlighted for each measurement. 9.3478 m + 2.73 m = What place value must the answer be reported to? 1. ones place 2. tenths place 3. hundredths place 4. thousandths place

Find an equation for a positive sine function having an amplitude of 1/4, a period of 4π, and a horizontal shift of 4π to the right. y = __/__ sin ( ___/___ x - π)

Find the average value gave of the function g on the given interval. g(t) = 2/(1 + t2')[0, 7]

Round to 2 significant figures. 42.598

An open-top rectangular box is being constructed to hold a volume of 350 in³. The base of the box is made from a material costing 6 cents/in². The front of the box must be decorated, and will cost 9 cents/in². The remainder of the sides will cost 3 cents/in². Find the dimensions that will minimize the cost of constructing this box. Front width: Depth: Height: in. in. in.

Round to 4 significant figures. 35.5450

Question 4 Given two vectors a = (1, 2, 1) and 6 = (4, 2k + 1, k) : a) Determine the value of k that makes them perpendicular b) Determine the value of k that makes them collinear / 2 marks /2 marks

corresponds between If y = cos x, what x-value to a y-value of and 2π?

Graph the polar equation. Identify the name of the shape. # 40 r 4 sin(50) a) Identify the shape: rose curve b) Identify the number of petals: 20 X c) Identify the length of each petal: 7 d) For what values of 0 within [0, 2x) mark the maximum value of r? e) Plot the tip of each petal below. (no need to connec them). 5 5 4 my 3 3 4 5

The volume of a cone of radius r and height h is given by V-rªh. tr²h. If the radius and the height V=- €½⁄³ centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? both increase at a constant rate of OA) 0.5*pi OB) 10*pi OC) 24*pi D) 54*pi E) 108*pi

The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? (A) f(1)<f'(1)<ƒ" (1) (B) f(1)<f'(1)<f'(1) (C) f'(1)<f(1)<ƒ^(1) (D) f'(1)<f(1)<f'(1) (E) f'(1)<f'(1)<ƒ(1)

Which sound device does the author use in the underlined words from the poem? In sunshine and in shadow, Had journeyed long, Singing a song, In search of Eldorado. A. repetition (repeated words) B. end rhyme (repeated rhyme at end of the line) Help Resources C. onomatopoeia (words that imitate sounds) D. alliteration (repeated beginning sounds)

Let P (x) represent the population (in thousands) in a region a years since year 1950. Use function notation to represent that the population is 540,000 in the year 1978. OP(28) = 540 O P(1978) = 540 OP(28) = 540,000 O P(1978) = 540,000

65. A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction?