Differentiation Questions and Answers

Suppose y = sin²x + 2sinx. Find dy/dx. Answer:

Consider the function f(x)=2x² - 6x² 90x+6 on the interval -4,9] Find the average or mean slope of the function on this interval By the Mean Value Theorem, we know there exists at least one e in the open interval (-4,9) such that f'(e) is equal to this mean slope Find all values of e that work and list them (separated by commas) in the box below. If there are no values of c that work, enter None List of numbers

The radius of a cylinder is increasing at a rate of 1 meter per hour, and the height of the clinder is decreasing at a rate of 4 meters per hour. At a certain instant, the base radius is 5 meters and the height is 8 meters. What is the rate of change of the volume of the cylinder at the instant? Recall that the volume of a cylinder is given by: V=πr²h B)-120 m³/hr C) 60 m³/hr D) - 80 m³/hr A) -60 m³/hr

Show that the function f(x)=x² +8x+2 has exactly one zero in the interval [-1.0]. Which theorem can be used to determine whether a function f(x) has any zeros in a given interval? A. Rolle's Theorem B. Extreme value theorem C. Intermediate value theorem D. Mean value theorem

The length of a rectangle is increasing at a rate of 9 cm/sec while the width w is decreasing at a rate of 9 cm/sec. When t= 8 cm and w= 15 cm, find the rates of change of the area, the perimeter, and the lengths of the diagonals of the rectangle. Determine which of these quantities are increasing, decreasing, or constant. The rate of change of the area of the rectangle is___cm² /sec. (Simplify your answer.)

A balloon, which always remains spherical, has a variable radius. If the radius is increasing at the rate of 3 units/sec then find the rate at which its volume is increasing with respect to time when its radius is 12 units Note that the volume of the sphere is given by V=4/3 π r^3 A) 48π units^3 / sec C) 576 π units^3 / sec D) 1728π units^3 / sec B) 144π units^3 / sec

Which sequence of transformations will yield the graph of g(x) = 2^x +8 + 1 Horizontal shift 1 units to the left; Vertical shift 8 units up Horizontal shift 8 units to the right; Vertical shift 1 units up Horizontal shift 1 units to the right; Vertical shift 8 units down Horizontal shift 8 units to the right; Vertical shift 1 units down Horizontal shift 8 units to the left; Vertical shift 1 units up

Select the best choice for the definition of average rate of change. A. The rate at which a function's y-values (output) are changing as compared to the function's x-values (input). B. The graph of a quadratic function that is a U-shaped curve. C. Is defined by an ordered pair and is considered the minimum point or the maximum point. D. A vertical line that passes through the vertex of the graph of a quadratic function.

The equation y = 664(1.031)t models the number of students in a certain state that go to out of state colleges/universities each year from 1992 through 1999, where y is the number of students that study out of state and t is the number of years after 1992. Round answers to the nearest whole student. Assuming that the above equation remains valid in the future, estimate 1) the number of students from above that studied out of state in 2002. 2) the number of students from above that studied out of state in 2018.

A fast food restaurant estimates that the mean sodium content in one of its breakfast sandwiches is 940 milligrams. A random sample of 44 breakfast sandwiches had a mean sodium content of 944 milligrams with a standard deviation of 18 milligrams. Is there sufficient evidence to say that the sodium content is more than 940 milligrams. Use a 1% level of significance. Conduct the appropriate hypothesis test.

Calculate the derivative of the following function. y = sin (4 cos x)

A theorem states that one local extremum implies absolute extremum. Verify that the following function satisfies the conditions of this theorem on its domain. Then find the location and value of the absolute extrema guaranteed by the theorem. f(x) = -3x² + 4x-6 The function has one local__ on its domain __ because f' changes sign from __to __ as x increases through the critical point, c.

Suppose that the total profit (in dollars) from the sale of x skateboards is given by P(x) = 30x - 0.6x² - 240 Determine the marginal average profit to approximate the profit from the sale of the 20th skateboard.

The functions=t³-15t² +75t,0≤t≤ 5, gives the position of a body moving on a coordinate line, with s in meters and t in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? a. What is the body's displacement for the given time interval? m (Simplify your answer.) What is the average velocity for the given time interval? m/s (Simplify your answer.) b. What is the body's speed at t=0? m/s (Simplify your answer.) *****

Calculate the derivative of y with respect to , if x³y + 3xy³ = x + y. dy/dx=

The radius of a sphere is measured as 9 centimeters, with a possible error of 0.025 centimeter. (a) Use differentials to approximate the possible propagated error, in cm³, in computing the volume of the sphere. error ± (b) Use differentials to approximate the possible propagated error, in cm2, in computing the surface area of the sphere. error + (c) Approximate the percent errors in parts (a) and (b). (Round your answers to two decimal places.) volume surface area

Calculate dcos(u)/dx for the following choices of u(x) : u(x) = 6 — x², dcos(u(x))/dx u(x) = 1/x, dcos(u(x))/dx u(x) = tan(x), dcos(u(x))/dx

Find a, b, c, d such that the cubic function f(x) = ax³ + bx² + cx+d has horizontal tangent lines at (0, -9) and (4,-41). a= b= c= d=

Find dy/dx by implicit differentiation. 9x3 + x²y-xy³ = 8

Find the equation of the line that is tangent to the graph of the given function at the specified point. y = x^4 - 5x^3 + 4x + 4; (1, 4) y=-11x - 7 y = -7x-3 y=-7x + 11 y = -11x + 15

Let f(x) = x + 64/x+6 (a) Find the domain of f. (Enter your answer using interval notation.) The domain of f is (b) Find the derivative of the function f. (c) Find the domain of f'. (Enter your answer using interval notation.) The domain of f'is (d) Determine the value(s) of x in the domain of f, if any, at which the function is not differentiable. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) (e) For what values of x does the graph of f have a horizontal tangent? (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Let f(x) =x³-6/x² + 1 (a) Find the derivative of the function f. (Simplify your answer completely.) f'(x) = (Express your answer as a single fraction.) (b) State the domain off and the domain of f'. (Enter your answer using interval notation.) The domain of fis The domain of f'is (c) Determine the value(s) of x in the domain of f, if any, at which the function is not differentiable. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Use the Product and Quotient Rules to find the derivative of each function. 1. f(x) = 3x² cos x 2. y = 7√x sinx 3. y secx tan x 4. f(x) = x²+2 / 2x-7 5. y = 3√x / x³-1 6. f(x) = x²-4 / cotx Evaluate the derivative of each function at the indicated point. 7. f(x) = (x³ - 3x) (2x² + 3x + 5) 8. f(x)= sin x / x 9. f(x) = tan x cotx

Businesses can buy multiple licenses for a data compression software at a total cost of approximately C(x) = 48x^2/3 dollars for x licenses. Find the derivative of this cost function at the following. (a) x=8 Interpret your answer. Each license costs this amount. When eight licenses are purchased, the average cost per license is this amount. Eight licenses cost this amount. When eight licenses are purchased, the cost to purchase the first license is approximately this amount. When eight licenses are purchased, the cost to purchase one more license is approximately this amount. (b) x = 64 Interpret your answer. Each license costs this amount. When sixty-four licenses are purchased, the cost to purchase one more license is approximately this amount. When sixty-four licenses are purchased, the average cost per license is this amount. Sixty-four licenses cost this amount. When sixty-four licenses are purchased, the cost to purchase the first license is approximately this amount.

Use the alternative form of the derivative to find the derivative at x = c (if it exists). (If the derivative does not exist at c, enter UNDEFINED.) f(x) = x³ +2x² +8, c = -2 f'(-2) =

For the function f(x) = 3x² + 2x - 6, evaluate and fully simplify each of the following. f(x + h) = f(x+h)-f(x)/h=

Give the average rate of change for f(x) = 2x² - 12x - 21 between x = 2 and x = 5. The average rate of change is

Find the difference quotient f(x+h)-f(x)/h for the function given below. f(x) = 9x - 1 Give a simplified expression involving x and h, if necessary. For example, if you found that the difference quotient was x + h, you would enter x + h.

A ball is thrown from a height of 156 feet with an initial downward velocity of 8 ft/s. The ball's height h (in feet) after t seconds is given by the following. h-156-81-16t^2 How long after the ball is thrown does it hit the ground?

Find the indefinite integral by making a change of variables. We are given the following indefinite integral and are told to make a change of variables to find the integral. It is generally best to use a substitution u = g(x) for the inner part of a composite function. In this case, there is a square root in the second factor. Therefore, we can set u to be the quantity under the square root. Find dU.

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. (Enter your answers using interval notation. If an answer cannot be expressed as an interval, enter EMPTY or .) f(x) = x² - 5x

Find the general solution of the differential equation and check the result by differentiation. dy/dt = 16t^7

Fill in the missing interpretations in the table using the example as a guide. A. The average rate of temperature change between to and toh hours after midnight. The instantaneous rate of temperature change to hours after midnight. B. The average rate of temperature change between to and to th hours after midnight. The instantaneous rate of temperature change at midnight. C. The average rate of temperature change between midnight and midnight D. The average rate of temperature change between midnight and midnight hours.

Suppose u and v are functions of x that are differentiable at x = 0 and that u(0) = 1, u'(0)=9, v(0) = -3, and v'(0) = 8. Find the values of the following derivatives at x = 0.

Find the derivative of the function by the limit process. g(t) = t³ + 3t

Find the two x-intercepts of the function f and show that f'(x) = 0 at some point between the two x-intercepts. f(x) = x(x - 9) (x, y) = (x, y) = (smaller x-value) X = (larger x-value) Find a value of x such that f'(x) = 0.

A person standing close to the edge on top of a 16-foot building throws a ball vertically upward. The quadratic function h(t) = - 16t² + 60t + 16 models the ball's height about the ground, h(t), in feet, t seconds after it was thrown. a) What is the maximum height of the ball? b) How many seconds does it take until the ball hits the ground?

Find f'(x) and f'(c). Function f(x) = (x³ + 2x)(4x4 + 4x − 3) f'(x) = f'(c) = Value of c c = 0

Consider the function on the interval (0, 2π). (a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) (b) Apply the First Derivative Test to identify the relative extrema.

Find the second derivative of the function. f(x) =cscx f"(x)=

Find the derivative of the function. f(t) = arccsc(-2t²) f'(t) =2/ t 4t4-1

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

Find any critical numbers of the function. (Enter your answers as a comma-separated list.) g(x) = x8 - 4x6

Find the derivative of the function. y = arctan x/2 - 1/2(x² + 4)

Verify that f has an inverse function. Then use the function f and the given real number a to find (f ¹)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.)

Use logarithmic differentiation to find dy/dx. y = (x + 1)(x - 6) / (x - 1)(x + 6)

Find dy/dx by implicit differentiation and evaluate the derivative at the given point. x³ + y³ = 16xy - 3, (8,5)

Find the derivative of the function. h(x) = x^2 arctan(5x)

The equation of the line tangent to the graph of f(x) = 5¹-x² at x = 1 is y = -2x + 1. y = -2x + 3. y=-2(In 5)x+ 1. y = -2(In 5)(x-1) + 1.

Find dy/dx at the given point for the equation. x = y³ - 9y² + 2, (-6, 1) Step 1 The given function is x = y3 - 9y² + 2. Differentiate the given function with respect to x. (3-9y² + 2) Step 2 To find