Differentiation Questions and Answers

dw dt For the given functions a express as a function of t both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t Then b evaluate w 6ye Inz x In 1 y tant z 1 1 dw at the given value of t dt by dw dt

40Write a 40 Write your answer as a fraction or a whole or

y sin In 5x 2x Different

Compute the following derivative using the method of your choice d cos x4 sin x dx d cos x4 sin x

Find the derivative of the function g x

ind the derivative of the function x g x 5x 3x

Find the following integral Note that you can check your answer by differentiatio S 6e4 y Vy dy

et dy dz 5x y e C

Find the derivative of the function f x below f x ln 15 e 9 f x

A fireworks mortar is launched straight upward from a pool deck platform 10 m off the ground at an initial velocity of 62 m sec The height of the mortar can be modeled by h t 4 9t 2 62t 10 where h t is the height in meters and t is the time in seconds after launch What is the maximum height Round to the nearest meter do not include units in your answer

Calculate the derivative of the following function y sin cos 5x dy dx 11 6 sin cos 2x cos 2x sin 2x

Calculate the derivative of the following function y e4x 2x 9 6 dy

15 Angle of Depression From a plane 3 000 feet in the air the angle of depression to the beginning of the runway is 3 50 What is the horizontal distance between the plane and runway That is if the plane flew at the same altitude how far would it be to the runway Give your answer to the nearest tenth of a mile 5 280 feet 1 mile

Calculate the derivative of the following function dy dx 1 y 5x 4 11 5 3 5x 1

Use a calculator to find the trigonometric value tan 37 tan 37 Type an integer or decimal rounded to four decimal places as needed

Write the following expression in terms of its cofunction K COS 3 The expression in terms of its cofunction is Type an exact answer in terms of Simplify your answer

Calculate the derivative of the following function dy dx 13 y 5x 4

A sequence is an infinite ordered list of numbers that is often defined by a function For example the sequence 2 4 6 8 is specified by the function f n 2n where n 1 2 3 The limit of such a sequence is lim f n provided the limit exists All the nx limit laws for limits at infinity may be applied to limits of sequences Find the limit of the following sequence or state that the limit does not exist 1 4 1 16 5 12 16 17 2 19 4 7 which is defined by f n n 15 for n 1 2 3 Select the correct choice below and if necessary fill in the answer box to complete your choice OA The limit of the sequence is B The limit does not exist and is neither o nor 00

Given the following figure 31 y 47 If h 12 find x and y 49 If y 10 find x and h r 43 h 48 If x 20 find h and y 50 If y 15 find r

uate the trigonometric function at the quadrantal angle or state that the expression is undefine 90 at is the value of csc 90 A 1 3 The answer is undefined C 1

If f x 4 cos 2 ln x find f x

f x f e f x

Find f x if x 4 f x In

Suppose that f 600 6000 and f 600 10 Estimate each of the following b f 600 5 c f 599 e f 599 75 a f 601 d f 598 a f 601 b f 600 5 6010 Simplify your answer Type an exact answer Simplify your answer Type an exact answer

Find the first derivative of each function below x x 1 x a y tan 3x sec 3x b f x c f x xe 1 et 1

Choose the appropriate description of the following function y 4sin 6x 7 Which of the following is the appropriate description of y 4sin 6x 7 1 OA amplitude 4 period phase shift 7 OB amplitude 7 period 3x phase shift 4 7 OC amplitude 4 period 3x phase shift 6 O D amplitude 4 period 3 phase shift 6

PROBLEM 6 For a function f R R f x xe 5x 3p find the right sided slant asymptote 5p examine the monotonicity and find local extreme values 2p verify if it is injective

y square root of 4 x 2 y 0

The formula 0 can be rewritten as 0 wt Using wt for 0 changes s r0 to s rot Use the formula s rot to find the value of the missing variable t T s 5 cm r 4 cm w radian per sec 8 t sec Type an integer or a fraction

Suppose that f 600 6000 and f 600 10 Estimate each of the following a f 601 b f 600 5 c f 599 d f 598 e f 599 75 a f 601 Simplify your answer Type an exact answer

Consider the cost function C x 7x2 12x 24 thousand dollars a What is the marginal cost at production level x 5 b Use the marginal cost at x 5 to estimate the cost of producing 5 50 units c Let R x x 41x 36 denote the revenue in thousands of dollars generated from the production of x units What is the break even point Recall that the break even point is when the revenue is equal to the cost d Compute and compare the marginal revenue and marginal cost at the break even point Should the company increase production beyond the break even point a The marginal cost at production level x 5 is C

An analysis of the daily output of a factory assembly line shows that about 70t t At t 6 the rate of production is units per hour 1 t3 units are produced after t hours of work 0 st 8 What is the rate of production in units per hour when t 62 108 CL

Question Let the curve C be the function f y 6y5 1 over the interval 1 y 7 Set up the integral for the arc length of the curve C Select the correct answer below 900y 1 dy T 1 30y dy 1 6y 2 dy 6y 1 1 dy of oy 1 dy GOLA 1 du

Given lim f x 7 and lim g x 4 evaluate 8118 lim 2f x 4g x 8118 lim 2 4f x 8118 lim x418 2f x 4g x 2 4f x 2 lim 2f x 4g x 2 4f x 8118 11

Suppose a vat contains 15 liters of water and 12 kg of salt At t 0 water is added at a rate of 20 liters minute while salt is added at a rate of 9 kg minute A function giving the concentration after t minutes will be C t express as a function of t As time passes the concentration will Decrease Increase to kg liter kg liter

PopQuiz 6 Prof Rudy Meangru Find the first derivative and do not simplify the Algebra y 1 e 1 e x

Find the values of a and b that make the function f x continuous everywhere 2x a x 3 f x ax 2b b 5x 3 x 3 x 3

Find g x for g x x 5 4

Find y if y 2x6 9x5 3 3

Use the washer method to find the volume of a shifted solu Question Define R as the region bounded above by the graph of f x 3x 1 below by the x axis on the left by x 1 and on t right by x 4 What is the volume of the solid obtained by rotating R about the line y 42 Enter an exact value in term of it Provide your answer below

ne position of a particle in the xy plane at time t is r t t 5 i t 4 j Find an equation in x and y whose graph is the path of the particle Then find the particle s velocity and acceleration vectors at t The equation for the path of the particle is y x 10x 29 The velocity vector at t 4 is v Simplify your answers UCLEB

Suppose lim f x 8 and lim g x 3 Find lim f x g x x 7 x 1 x 1 I 1

Estimate the limit numerically or state that the limit does not exist 9 1 lim 0 X Give your answer to at least three decimal places xample

The equation r t cos 2t i sin 2t j t20 describes the motion of a particle moving along the unit circle Answer the following questions about the behavior of the particle a Does the particle have constant speed If so what is its constant speed b Is the particle s acceleration vector always orthogonal to its velocity vector c Does the particle move clockwise or counterclockwise around the circle d Does the particle begin at the point 1 0 a Select the correct choice below and fill in any answer boxes within your choice OA The particle s constant speed is OB The particle s speed is not constant

The equation r t 3t 1 8t 6 j 4t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product of its speed and direction What is the velocity vector v t i j k GEL

Problems 27 34 refer to right triangle ABC with C 90 In each case solve for all the missing parts using the given information 27 A 20 c 24 in 28 A 40 c 36 in 29 B 31 a 76 c 5 8 mi 35 m b 97 m 33 b 150 cm c 200 cm 30 B 21 c 4 2 mi 32 a 78 m b 83 m 34 b 320 cm c 650 cm

a For what values of x is the function f x 1x 251 differentiable Enter your answer using interval notation X X Find a formula for f f x b Sketch the graph of f 20 10 40 X 20 X f x 30 20 Sketch the graph of f 10 10 30 if x if x f x 40 20 20 40 10 20 20 X X 40 10 40 f x 20 WM 10 10 10 10 f x 40 20 20 40 20 40 20 f x 10 10 10 f x 20 10 10 20 5 10 10 20 X 20 f x 30 VAV 10 20 10 10 10 20 30 f x 20 10 10 20 10 10 20 20 9

Find the first and second derivat y x 8 y y 1 2 x 8

Use a graph off to estimate lim f x or to show that the limit does not exist Evaluate f x near x a to support you X 8 conjecture f x 1 cos 5x 5 5 x 1 Each graph is displayed in a 1 3 by 0 5 window Use the graphing utility to estimate lim f x Select the correct choice below and if necessary fill in the answer box X 1 complete your choice a 1 A The limit appears to be approximately 2 5 Round to the nearest tenth as needed B The limit does not exist X b Evaluate f x for values of x near 1 to support your conjecture 0 9 0 99 0 999 Round to six decimal places as needed 1 001 1 01 1 1

The third derivative of a function f x is the derivative of the second derivative f x and is denoted by f x Compute f x for the following function f x 5x5 4x4 7x f x