Differentiation Questions and Answers

The concentration of a medicine in the bloodstream after t hours of taking it is giver 0 1t t 3 where t is the elapsed time in hours measured from by the equation C t 11 00 am How much is in the bloodstream at 11 00 am 0 006 25 0 0053 0

Use the graph off to find the largest open interval on which is increasing and the largest open interval on which is decreasing Enter your answers using interval notation y N 2 N a Find the largest open interval s on which is increasing If two open intervals are equally large enter your answer as a comma separated list of intervals b Find the largest open interval s on which fis decreasing If two open intervals are equally large enter your answer as a comma separated list of intervals

Evaluate the trigonometric function COS

Suppose you deposit 2000 at 3 interest compounded continously Find the average value of you account during the first 4 years A

d g 22 Let g z Determine the third derivative of g dz3 10 cos z e 10 sin z 3 10 sin 2 10 cos z e

4 pts If f x y ey 5y x then fry ry A 1 xy ey 10y B yery 5y C y ery Q D E xexy 10xy ey 10 x

Find the value of the derivative if it exists at the given extremum If an answer does not exist enter DNE f x x 5 2 3 f 5 10 8 6 5 0 2 2 2

gives the height as time passes of a typical pine tree grown for lumber at a managed site Tree age years 42 49 14 21 28 35 64 72 79 84 Height feet If H t is the height of the tree after t years construct a table of estimated values for H Use a one sided difference quotient with an adjacent point for the first and last values and the average of two difference quotients with adjacent points for all other values Round your answers to two decimal places 2 0 1 5 1 0 t 0 5 14 21 28 35 42 49 40 54 64 Sketch the graph of H H 72 79 84 H t 40 x X X x x X 54 H 2 0 1 5 1 0 0 5 H 2 0 1 5 1 0 0 5 2 0 H 1 5 1 0 0 5

f x 2 x 0 X 11 X X 1 0 3 1 R D

Suppose and y are both differentiable functions of t and Determine dx dx dt y 4x 2x 2 dy given 8 and x 4 dt

Let f y arctan y Determine the derivative of f f y Determine the slope of f at y 7 7

How many periods of the function y sec x are there between 7 and JU 5 211 000 periods Enter your answer as a decimal Ente

13 g t 3 sin 5t 2test 2t 2 14 Find dr

3 7 f x 2 2 2 2 3x 2 Answer f x 12 2 x 3x 2 4 8 f x e3x 2x tan x 3 3x 9 f x 4xe 1 cosx

Find the derivative of each function dx 1 f x x 2 f x 3e 3 f x 4 2 f x x

12 g x 13 h t 1 tan x 1 tan x 1 cost

6 f x 3x cotx 7 g x 2x csc x 8 h x 1 sin x 1 sin x

4 f x 2 cos x 3 4 X x 2 5 g x 3 sin x 2 6 f x 3x cotx

dy 9 Find the derivative at of y 4te 21

Let g x Use logarithmic differentiation to determine the derivative of g d dx g x x 9 5 x 8 6 x 1 x 2

Which of the following represents z 6 3 6j in trigonometric form

10 points If f 3 8 f 3 4 g 3 2 g 3 6 find F 3 if F x f x g x

f x 4x f x 4x limf x h f x Find also the equation of the tangent line to the graph of the h 0 h function at 14 12 3 Find the derivative of f x using the definition f x lim 4x 0

75 1072 31 3 12 3 3 3 4 3 5 42

y 14 y 14 Use logarithmic differentiation to determine the derivative of f Let f y Dy f y

Consider the parabola y 8x x a Find the slope of the tangent line to the parabola at the point 1 7 b Find an equation of the tangent line in part a y c Graph the parabola and the tangent line y 20 10 20 10 20 10 20 X 00 20 10 y 20 10 10 10 20 10 20 10 y 20 10 10 201 10 20 X 00 20 10 y 20 10 10 201

8 5 4 3 2 2 4 Based on the graph above estimate to one decimal place the average rate of change from x 5 Question Help Video Message instructor

Let h v 5v7v 12 Use logarithmic differentiation to determine the derivative of h Dv h v

Let g w sin21w 11w Use logarithmic differentiation to determine the derivative of g d 7 g w

12 Consider the shaded region bounded by the sine and cosine functions depicted below YA y cos x 0 5 0 5 0 5 y sin x 0 5 1 X Use the washer method to find the volume of the solid of revolution formed by revolving the shaded region around the r axis Hint Recall the trigonometric identity cos r sin r cos 2 7 Take special care in finding the upper limit of integration

4 Ox 2 Ox 0 ch 0 x 2 Ox 2 2 64 Over what interval is f x concave down y f x 5

10 points Show that f x 2 x 2 x 1 2x3 x 1 is continuous 5 pts and differentiable 5 pts at x 1

This question asks for a number or a calculation You can enter whole numbers negative numbers or decimal numbers If you enter a decimal value be sure to give at least 3 decimal places Alternatively you can enter mathematical expressions Some examples Enter To get sqrt 4 2 5 3 Enter 5 V 2 Enter 3 2 3 9 log 100 log 100 2 Enter 4 2 2 Note that for functions like log and sqrt you must use parentheses log 100 is good but log100 and log 100 will not work Try it here Use the Preview button to see how the computer is interpreting what you typed 7100 5 3 3 1 2 5

Let h t In 4 et 4 et Determine the derivative of h h t

Find the following using the table below X 1 2 3 4 f x 1 2 3 4 f x 3 1 4 2 g x 3 4 2 1 g x 1 3 4 2 h 2 if h x f x g x h 2 if h x H g x h 2 if h x f g x

Graph the following function over the indicated interval y 5 e x 4 2 Choose the correct graph below OA G OB Ay O C O D

8 2 7 Determine the derivative of g Let g z dg

dy Find dx if y 5x cosx A B C D cosx 5x cosx 5x 2 cotx In cosx 2 1 x tanx Incesx X 2 5x2sinx cosx 1 5 cosx 2 X sx sinx 2 1 5 cosx 1 N X

Let g z In 8 A 6 e 6 e Determine the derivative of g

A sin kx or y A cos kx whose graph matches the function shown Find a function of the form y below AA 8 7 6 5 3 2 2 Y 1 Leave your answer in exact form if necessary type pi for 1 2 3

a Find two explicit functions by solving the equation for y in terms of x Y1 9 1 V9 positive function Y2 9 1 b Sketch the graph of the equation and label the parts given by the corresponding explicit functions y y dy dx 6 negative function 6 4 2 H 4 X 6 2 21 2 4 y 2 c Differentiate the explicit functions dy dx 2 2 d Find dy dx through implicit differentiation 2 4 4 X 6 6 6 4 6 2 4 6 2 25 H 2 2 2 4 2 4 4

Let f w 2w 3 Determine the derivative of f df

Find a function of the form y function shown below 4 3 2 An 1 1 2 3 A sin kx Cor y A cos kx C whose graph matches the 6 y 4 54 Leave your answer in exact form if necessary type pi for 4

Consider Determine the derivative of y with respect to x dy dx x 6 y 5 5

Consider 5ae 7y 17ye5 Determine the derivative of y with respect to x y

Let g u 7u log7 Determine the derivative of g da

Let h u 6 4u 8u 7 Determine the derivative of h h u Determine the slope of h at u 5 h 5

6 Describe the x values at which the graph of the function f x 2 2 4 2 6 4 6 X given below is differe

With these types of questions be sure to use the variables indicated In your answer you can also use mathematical expressions for numerical calculations Examples Type 3x 2 5 2 x 3 x sqrt x 5 3 x 7 1 x x 1 5 3x 2 3 Your input To get 3x 5 Your input 2 x 3 x x 5 32 7 1 x x 1 5 2 x 3 3 B Enter the expression A Notice the symbol for powers Notice you need parentheses around the numerator and denominator Be sure to put parentheses around the input sqrt x 3 is OK but sqrtx 3 sqrt x 3 is not Parentheses keeps the 7 together as the power You may need nested parentheses Unless the problem gives specific directions any algebraically equivalent expression is acceptable For example if the answer was x 2 x 3 then x 3 x 2 and 3 x 2 x would also be acceptable Try it here Use the Preview button to see how the computer is interpreting what you typed A Enter the expression 4x 1a 8 Comment Fractions are automatically formatted vertically when previewed 2y 3 4y C Enter the expression w 4 w 1

Indicate whether the graph specifies a function Q 10 Ay 10 10 X 10 O N O The graph does specify a function O The graph does not specify a function