Differentiation Questions and Answers

For what value s of a is the tangent line to the graph of parallel to the line y 2x 0 4 f x 4x 6x 142x 48 If there is more than one value of a list them as a comma separated list

Let f x x 8 Evaluate the following limits T lim Z 8 lim f x f 8 x 8 CAMENTE f x 8 x 8 Thus the function 2 P ENCO

Find the points on the graph of f x 2x 2x 3 List the x values of these points I value s Separate answers by commas if ther 6x 6x 90x 24 where the tangent is horizontal

graph by using the vertical line test If it is use the graph to find a The domain and range b The intercepts if any c Any symmetry with respect to the x axis the y axis or the origin IYU A The domain is 00 The range is Type your answers in interval notation C 18 X 10 12 a If the graph is that of a function what are its domain and range Select the correct choice below and h if necessary fill in the answer box to complete your choice

3V 33 The radius r in inches of a basketball is related to the volume V by r V 4 into the basketball so the volume after t seconds is given by V t 24 Find the composite function V 1 a 25 Find the time when the radius reaches 9 inches Air is pumped

Find the following for the function f x x 14 b f 3 a f 0 e f x f f x 3 c f 3 g f 7x a f 0 14 Simplify your answer Type an integer or a fraction b f 3 Simplify your answer Type an integer or a fraction d f x h f x h

Given that a 11 630 g milk chocolate bar contains 7 815 g of sugar calculate the percentage of sugar present in 11 630 g of milk chocolate bar keeping in mind that the answer should have four significant figures two decimal places Express your answer numerically to four significant figures View Available Hint a

7 y 2x 2 8 A 37 100 B 2 1 3 C D

Let f x arsitate 3x57x4 7z4 3 Then

Let x 5 x x 4 x 6 Then TOETUST 3

If f t 7 t f t 9 vt find f t

If f u 2u 5u find f u

Differentiate the following function V r V r 43 7

Let f x 2x8 525 3x 5x find f x Answer Find f 1

Suppose f x y is a differentiable function satisfying f xo yo 10 and fy 2o yo 5 Then the directional derivative of f at xo yo in the direction of the vector v 4 3 is 10 5 55 9

Let r t x t y t and z f x y f r t Which of the following dz expressions are equal to Vf r t r t D f r Select all that apply dt f x t y t f x t y t of dx of dy x dt dy dt f r t r t r t

Identify the radicand x y y 5 The radicand is

Use a calculator to approximate the square root 209 49 209 49 Round to the nearest thousa

The displacement d in mm of a piano wire as a function of the time t in s is given by d 2 0 sin 178t cos 178t How fast is the displacement changing when t 3 0 ms The change in displacement is mm s when t 3 0 ms Do not round until the final answer Then round to the nearest whole number as needed

Power P is the time rate of change of work W Find the equation for the power in a circuit for which W 6 sin 7t

For a certain electric circuit the current in A is given by i 1 68e 66 1t sin 194t where t is the time in s Find di dt the expression for b E m

Suppose that x and y are related by the equation 4x 3y 5 and use implicit differentiation to determine dy dx 11 dy dx

The energy E in J dissipated by a certain resistor after t seconds is given by E In t 1 0 5t At what time is the energy dissipated the greatest t second s

Find the derivative of the given function dR dt 8 R 4 tan 3t

Evaluate the following indefinite integral 4x 3 3 dx 4x dx 3 3

Let s represents the displacement and let t represents the time for an object moving with rectilinear motion according to the given function Find the instantaneous velocity for the given time s 60 180t 60t t 1 5 The instantaneous velocity is Simplify your answer

Find the radius of gyration of a plate covering the region bounded by y 2x x 2 and the x axis with respect to the y axis R

b The length of arc s of a curve from x a to x b is given by the formula s a 3 2 described by the equation y 0 16x 1 dx 2 dx The cable of a bridge can be from x 0 to x 100 ft Find the length of the cable S feet Round the final answer to the nearest integer as needed Round all intermediate values to the nearest ten thousandth as needed 100 ft

The graph of fis translated a whole number of units horizontally and vertically to obtain the graph of g The function fis defined by f x x Write down the expression for g x g x 0 2 0101 X G E

TU Find the volume generated by revolving the region bounded by y sec x x 0 x and y 0 about the x ax V Type an exact answer using as needed

Determine the equation of the circle graphed below 12 11 10 96 8 76 7 6 54321 4 12 11 10 9 8 7 6 5 4 3 2 1 1 2 co 4 6 7 8 50 y 5 2 1 2 3 4 5 6 7 8 9 10 11 12 4 3

An element with mass 490 grams decays by 28 6 per minute How much of the element is remaining after 16 minutes to the nearest 10th of a gram

ng along a gawa the relationship between distance and time Answer Attempt 1 out of 2 The independent variable x represents the variable is the on the So C 3 50 miles per hour Answer the questions below regarding V A function relating these variables is C x meaning after 3 because the and the dependent 2 number of miles traveled number of hours driving depends

Find the point on the graph of z 3x 3y2 at which vector n 12 18 1 is normal to the tangent plane P 2 3 14 A

Consider the partial derivatives Is there a function f which has these partial derivatives Yes If so what is it f 12x y Enter none if there is no such function any others Are there Yes y fz x y 5x y 6x y fy x y 4x5y 2x

z ze Find 02 0u and 02 0e The variables are restricted to domains on which the functions are defined 02 0u 02 00 z u v u v

z x y e find az ou and az av The variables are restricted to domains on which the functions are defined z du 4 z dv x U y ln v

1 x 3 x 5 Determine if the point 2 25 is on the graph of the function f x x Is the point on the graph of f x O Yes O No

Differentiate d dx y 6x 4 6x 4

Differentiate y 24 3 x x 24 dx 3 x x

Use the Chain Rule to calculate the partial derivatives g x y cos x y x u 4v y u v 8g du 94 ag du Express your answer in terms of the independent variables u v

Let f x 4 X a Find the derivative f x b Find f 2 a What is the first step in finding the derivative of f x O A Rewrite the function in the form f x OB Substitute x h for x in the original function OC Rewrite the function in the form f x rx OD Rewrite the function in the form f x x 1 Rewrite f x 4 X in this form f x 0 Which rule of derivatives should be used to differentiate functions of this form OA The rule that if f x ax then f x a OBThe rule that if f x mx b then f x m

Consider the partial derivatives s there a function f which has these partial derivatives Yes If so what is it Enter none if there is no such function Are there others any Yes v fz x y 5x y 6x y fy x y 4x y 22

10 Domain none 4 5 7 7 4 5 4 5 5 X 5 S 10 4 The given curve is the graph of a function of x by the Vertical Line Test The given curve is not the graph of a function of x by the Vertical Line Test 5 10 X

If z sin and x ln u and y v find the following partial derivatives using the chain rule Enter your answers as functions of u and v z

8g Use the Chain Rule to evaluate the partial derivative at the point r 0 2 2 4 80 where g x y 3 8g 80 r 0 2 2 1 8x 6y 11 x r sin 0 y r cos 0 7

Let f x y z x y z and x s t y s t and z s t a Calculate the primary derivatives of 32 dx 4x y dy 2x4y 38 38 af af 8z dx s b Calculate dy s dz as c Use the Chain Rule to compute af ds In eller 3

Find an equation of the tangent plane to the surface z 3y 5x at the point 3 1 42 2

Find the second partial derivatives of the function f x y x y 5x y a f x 1 2 3 4 8 f y x 8 f dxdy a f dy Chalk

Calculate all four second order partial derivatives and check that fay fyr Assume the variables are restricted to a domain on which the function is defined f x y 4 sin 2x cos 6y fyy 8 9