Differentiation Questions and Answers
Calculus
DifferentiationFor 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
Calculus
DifferentiationLet 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
Calculus
DifferentiationFind 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
Calculus
Differentiationgraph 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
Calculus
Differentiation3V 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
Calculus
DifferentiationFind 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
Calculus
DifferentiationGiven 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
Calculus
DifferentiationSuppose 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
Calculus
DifferentiationLet 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
Calculus
DifferentiationUse a calculator to approximate the square root 209 49 209 49 Round to the nearest thousa
Calculus
DifferentiationThe 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
Calculus
DifferentiationPower 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
Calculus
DifferentiationFor 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
Calculus
DifferentiationSuppose that x and y are related by the equation 4x 3y 5 and use implicit differentiation to determine dy dx 11 dy dx
Calculus
DifferentiationThe 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
Calculus
DifferentiationLet 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
Calculus
DifferentiationFind 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
Calculus
Differentiationb 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
Calculus
DifferentiationThe 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
Calculus
DifferentiationTU 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
Calculus
DifferentiationDetermine 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
Calculus
DifferentiationAn 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
Calculus
Differentiationng 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
Calculus
DifferentiationFind 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
Calculus
DifferentiationConsider 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
Calculus
Differentiationz 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
Calculus
Differentiationz 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
Calculus
Differentiation1 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
Calculus
DifferentiationUse 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
Calculus
DifferentiationLet 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
Calculus
DifferentiationConsider 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
Calculus
Differentiation10 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
Calculus
DifferentiationIf 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
Calculus
Differentiation8g 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
Calculus
DifferentiationLet 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
Calculus
DifferentiationFind an equation of the tangent plane to the surface z 3y 5x at the point 3 1 42 2
Calculus
DifferentiationFind 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
Calculus
DifferentiationCalculate 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