Differentiation Questions and Answers

Calculate all four second order partial derivatives of f x y 3x y 7xy fax x y fry x y fyr x y fyy x y I 9 9

Find the partial derivative indicated Assume the variables are restricted to a domain on which the function is defined z sin 7x y 4xy

Suppose f x y xy 3 Compute the following values f 5 4 f 4 5 f 0 0 f 2 4 f x 5 f 4 y f t 4t f uv u v 12 8

Find the equation of a line tangent to the curve xy xy x 1 at the point 1 2 OA y 5 667 2 667x OB y 2 x OC y 2 667 1 333x OD y 2 667 1 667x O E y 5 3x

If q is positive and increasing for what value of q is the rate of increase of q twelve times that of q A 3 OB 4 5 OC 2 OD 12 OE 36

4 12 points Let f r be a scalar valued function of a single variable r satisfying f 0 df Define a function dr 2 f 0 1 where f x g x y cos x y f x Find g 0 3 0 and 9 0 3 0 3

Find the first and second derivatives d dx y 29 x 29 x d 2 dx 29 x

Differentiate y X x 2 2 x dx 2 11

Let f x 7 X a Find the derivative f x b Find f 2 1 a What is the first step in finding the derivative of f x 7 A Substitute x h for x in the original function OB Rewrite the function in the form f x rx OC Rewrite the function in the form f x x O D Rewrite the function in the form f x Rewrite f x 1 7 X in this form XIL X f x Which rule of derivatives should be used to differentiate functions of this form A The rule that if f x mx b then f x m

Determine whether the following limit exists Compute the limit if it exists x 3x 4 X 4 lim X 4 Is the function f x x 3x 4 X 4 OYes O No Does the function need to be defined at x 4 in order for its limit to exist at x 4 defined when x 4 O No OYes To evaluate the limit begin by factoring the numerator x 3x 4 x 3x 4 X 4 Factor completely Do not simplify When f x is defined common binomial factors can be cancelled from the numerator and denominator 11 Simplify your answer

Factor the following polynomial 4x 32x 64 4x 32x 64

a Find the slope of the tangent to the curve y 2 x at the point where x a b Find equations of the tangent lines at the points 1 2 and 9 6 c Graph the curve and both tangents on a common screen

A particle moves along a line so that at time t where 0 t its position t is given by s t 4 sint 10 What is the acceleration of the 2 particle the first time its velocity equals zero A 3 969 OB 0 745 C 2 550 D 1 323 E 5 197

Which of the following will not give the instantaneous rate of change of f x with respect to x when x 3 f 3 f 3 0 1 0 1 O A OB lim x 3 x 3 OC lim h 0 D lim I 0 f x f 3 OE lim h 0 f 3 h f 3 h f 3 f x 3 x f 3 h f 3 h 2h

A 17 m Using trapezoids estimate the total distance the runner travels from t 0 to t 6 seconds B 12 m C 24 m D 26 m t seconds 0 2 46 2 5 8 12 E 40 m v t m sec

A puppy weighs 2 0 pounds at birth and 3 5 pounds two months later If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight then how much will the puppy weigh when it is 3 months old OA 4 2 pounds OB 5 6 pounds C 6 5 pounds D 4 6 pounds E 4 8 pounds

The velocity of a particle is shown in the graph below For which x values would the acceleration of the particle be zero Click here for long description OA x 22 X B X 1 OC x 1 and x 2 OD x 0 and x 1 4 OE x 2

The graph of y is concave downward for all values of x such that A x 2 OB x 0 O C x 0 OD x 2 OE x 5

If fis a differentiable function f a is given by which of the following lim 0 f a h f a h f x f a limx a x a limx a f x h f x h OA I only OB II only OC I and II only OD I and III only E I II and III

The region enclosed by the curve y ex the x axis and the lines x 0 and x 1 is revolved about the x axis Find the volume of the resulting solid formed A 2 1 T B 3T O C T 2 O D e 1 2 E e

The tank s base diameter is 40 m and a height of 10 m A Write an expression for the rate of change of the water level with respect to time in terms of h the water s height in the tank h B Assume that at t 0 the tank of water is empty Find the water level h as a function of the time t

The volume of the solid obtained by revolving the region enclosed by the ellipse x 9y2 9 about the x axis is OA 2 OB 4T OC 9T OD 6TT E 12T

Which of the following differential equations with initial condition yields a solution curve that exhibits modified exponential growth such that y goes to negative infinity as t goes to positive infinity Hint Check each one by solving the differential equation quickly Remember to solve the differential equation the quick and easy way recall the fact that the solution to the differential equation k y a is y Aekt Aekt a So you just need to write the differential equation in the proper form first Then of course you must solve for the constant A using the initial condition OA 2 y 200 with initial condition y 0 B 38 38 100 D dy dt 2 y 200 with initial condition y 0 50 dy O C c 3 5 y 123 with initial condition y 0 100 dy 3 5 y 123 with initial condition y 0 50 dt E 2 y 200 with initial condition y 0 50

For the position function s t instantaneous velocity at t 0 Time Interval Average Velocity 0 1 Complete the following table Time Interval 18 t 1 0 1 complete the following table with the appropriate average velocities Then make a conjecture about the value of th 0 0 5 0 0 5 0 0 1 12 0 0 1 0 0 01 0 0 01 Average 9 16 4 Velocity Type an integer or decimal rounded to three decimal places as needed 17 8 0 0 001 0 0 001 17 9

The change in the momentum of an object Ap is given by the force F acting on the object multiplied by the time interval that the force was acting Ap FAt If the force in newtons acting on a particular object is given by F t cost what s the total change in momentum of the object from time t 5 until t 7 seconds A 0 708 newton sec B 0 909 newton sec C 1 616 newtons sec D 0 402 newton sec E 1 416 newtons sec

If y5 3x y 5x4 49 then at the point 1 2 is dy dx O A 11 23 OB 11 23 O C 1 O D OE O 10 3

Suppose a certain country s population has constant relative birth and death rates of 97 births per thousand people per year and 47 deaths per thousand people per year respectively Assume also that approximately 30000 people emigrate leave from the country every year Which of the following equations best models the population P P t of the country where tis in years O A 50P 30000 dP dt dP B dt 05P 30000 dP O C 5P 30000 dt OD O E d 05P 30000 5P 30

A rock is dropped off the edge of a cliff and its distances in feet from the top of the cliff after t seconds is s t 16t Assume the distance from the top of the cliff to the water below is 256 ft Complete parts a and b below a When will the rock strike the water The rock will strike the water at seconds Simplify your answer

g x x3 2x 1 What is the slope of the tangent line to the curve g x at the point 2 1 O A 1 OB 5 O C 8 OD 14 O E

A 25 ft ladder is leaning against a vertical wall At what rate with respect to time is the angle between the ground and the ladder changing if the top of the ladder is sliding down the wall at the rate of rinches per second at the moment that the top of the ladder is h feet from the ground You re looking here for an equation in terms of h and 0 A 25 1 4 O A OB O C E 25 1 13 300 A 25 ft 12 1 4 1 25 1 4

Suppose that f x y is a differentiable function Which of the following are correct statements Select all that apply fx x y lim h 0 O O fy x y f x y fy x y lim h 0 f x h y f x y h f x y lim f x y h f x y h f x h y h f x y h

Consider the function f x 20x e on the domain U On its domain the curve y f x A attains its maximum value at x value at x 0 OB attains its maximum value at x minimum value O C attains its maximum value at x value at x 0 OD attains its maximum value at x value OE attains its maximum value at x minimum value and attains its minimum and does not have a and attains its minimum and does have a minimum and does not have a

Which of the following is the correct formula for the slope of the tangent line OA Mtan lim OB Mtan lim f t f a t a OD Mtan lim f b f t b t OC Mtan lim t a f t f a t a f b f t b t

What is the slope of the secant line between the points a f a and b f b on the graph of f Which of the following is the correct formula for the slope of the secant line OA Msec OB Msec OC Msec OD Msec f b f a b a f b f a b a f a f b b a f b f a b a

The position of a particle moving in a straight line is given by s t et cos 5t for t 0 where tis in seconds If the particle changes direction at time T seconds then Tmust satisfy the equation A 5e t sin 5t 0 OB tan 57 3 OC cos 5T 0 O D 5T arctan 5 E cot 57 5

dr VI dx evit got O A O B 3r e 1 e x 2 e r 2 1 e r 1 2 OC 1 1 2 c O D e3x 3 2 1 e z E 3 1 e a 1 2

dlog5 x 1 dx O A OB B O C Hol O D 1 2 x 1 In 5 2 2 1 In 5 x 2 x 1 X D 2 In 5 2 1 OE None of the above

f x ln 3x 36 Then f x 24x O A 3 36 OB O C O D 12x In 3x 36 3 x 36 12x 37 36 2 ln 3x 36 3 36 6x O E E 3 36

Suppose f x 9 x Then on any interval where the inverse function y f x exists d x dx OA 2x 1 B 9 x 1 2 O c 2 9 x 2 OD 9 x OE 2x

Suppose that y f x x 4x 4 Then on any interval where the inverse function y f x exists the derivative of y f x with respect to x is Hint x 4x 4 can be factored and rewritten as something squared A Both where x and y satisfy the equation y x 4x 4 and x 1 2 B 1 2y 4 1 2x 4 OC 2 4 where x and y satisfy the equation y x 4x 4 2y OD 2 O Ex 1 2

Find the derivative of the given function r t tan 2 5t 11

Suppose that x and y are related by the equation 4x y 4 and use implicit differentiation to determine dy dx dy

Find the derivative of the given function y 8sin 4x 7 dy dx II

Linearize f x 4 x for a 3 and use it to approximate the value of 1 8 x

A Find the equation of the normal line to the curve In y3 2x 1 at the point 1 1 I

Find a parameterization for the circle of radius 4 in the xy plane centered at the origin clockwise r t y t z t 9

customer is using while a kilowatt hour is one kilowatt of power being used for one hour For constant power use the number of kilowatt hours used is calculated by kilowatt hours kilowatts time in hours Thus if customers use 5 kilowatts for 30 minutes they ll have used 5 kilowatts hours 2 5 kilowatt hours Suppose the power use of a customer over a 30 day period is given by the continuous function P f t where P is kilowatts is time in hours and 0 corresponds to the beginning of the 30 day period A Approximate with a Riemann sum the total number of kilowatt hours used by the customer in the 30 day period and explain why your Riemann sum is an approximation of the desired property B Derive an expression representing the total number of kilowatt hours used by the customer in the 30 day period and explain your reasoning This expression should not be an approximation C Consider the following table of data for the function 1 f 1 0 2 3 2 5 2 1 3 9 3 6 5 5 4 5 7 5 6 8 1 2 9 1 0 10 1 8 3 4 I Recall that r represents the number of kilowatts being used by a customer at timer hours

1 Consider the function f x x e 2x for x 0 where n is some unknown constant greater than 2 A Find the constant for which the function f x attains its maximum value at x 3 Make certain to justify that a maximum is attained I

A Find the equation of the normal line to the curve In y 2x 1 at the point 1 1 1

3 The position of a particle in rectilinear motion is given by s t tln for t 0 where is in seconds Find the time 0 at which the velocity of the particle is 0 and find the particle s acceleration at that time as well You must find exact values don t approximate with a calculator