Differentiation Questions and Answers

Evaluate the derivative of the following function f x 3x x 3x f x 3x 3 log 3x 3 Simplify your answer

Use logarithmic differentiation to evaluate f x f x f x x 3 4x 8 1

Use logarithmic differentiation to evaluate f x f x f x 5 2 3 5 x 6 x 8 5x 1 36

5 Evaluate the derivative of the following function f x 3x x f x Simplify your answer

Show that d dx d d dx Inkx Compute the derivative of the natural log of kx Begin by letting u kx and substituting u into the equation d In kx In u dx Compute the derivative using the chain rule In u dx d 1 d u dx In x given that x 0 and k 0 is a real number dx u Substitute kx back in for u where it appears in the equation RILE x In kx x kx

3 d 1 Use x e to explain why In x for x 0 dx X First explain why x must be greater than zero What is the range of the function f y e A 0 00 B 0 00 C 0 00 D 1 00 Which of the following must be true about the value of x if x e A If y is negative then x is also negative B xze C X20 D X 0 explain why In x dx X Begin by solving the equation x e for y

d Show that Inkx d Compute the derivative of the natural log of kx Begin by letting u kx and substituting u into the equation In kx d d dx In x given that x 0 and k 0 is a real number dx

K Use the following table to determine the indicated derivatives or state that the derivative cannot be determined 6 0 1 5 3 4 3 6 8 x f x 7 1 3 f x 2 3 2 3 2 a f 4 b f 1 c 3 d f 3 a Choose the correct answer below and if necessary fill in the answer box within your choice A 4 3 Type an integer or a fraction OB 4 cannot be determined b Choose the correct answer below and if necessary fill in the answer box within your choice OA 1 cannot be determined OB 1 Type an integer or a fraction

O Points 0 of 2 How many years will it take for an initial investment of 20 000 to grow to 70 000 Assume a rate of interest of 2 compounded continuously t will take about 27 20 years for the investment to grow to 70 000 Round to two decimal places as needed

Jerome will be buying a used car for 10 000 in 3 years How much money should he ask his parents for now so that if he invests it at 10 compounded continuously he will have enough to buy the car Jerome should ask for Round to the nearest cent as needed www Save mou

Find the principal needed now to get the given amount that is find the present value To get 2000 after 2 years at 7 compounded daily 3 4 The present value of 2000 is Round to the nearest cent as needed

K Find the inverse of the following function and write it in the form f x 3 f x 2 X 1 for x 0 The inverse function is f x 3 xw 1 for 0 x 3 BOWWALI

Find the derivative of the inverse of the following function at the specified point on the graph of the inverse function You do not need to find f f x 4x 2 14 4 The derivative is

K Find all the inverses associated with the following function and state their domains 22x x 22 f x 22x The inverse s is are f x 22 Type an exact answer using radicals as needed Use a comma to separate answers as needed The domain of each inverse is Type your answer in interval notation CITS K

7 The number of students S infected by the flu at Gaither High School is given by the function 1600 where t is measured in days s t 1 99e 0 4t a How many students were infected with the flu at t 0 1600 1600 1600 1 99 180 S 0 1 99e 0 6 4 b How many students will be infected with the flu after 5 days 1600 8 5 1600 1600 1 99 04 5 16 398 996 2 1 99 13 398 1644 lim8 t 1600 X D Find lim S t and interpret its meaning in the context of the problem x 00 1 111 126 r

Sinay tas 7 Find the equation of the tangent line using the point ysin 8x x os 34

A basebasll diamond is a square with side 90 ft A batter hits the ball and runs toward first base with a speed of 24 ft s 90 ft a At what rate is his distance from second base decreasing when he is halfway to first base Round your answer to one decimal place b At what rate is his distance from third base increasing at the same moment Round your answer to one decimal place ft s Enter an integer or decimal number more Question Help Message incr

When air expands adiabatically without gaining or losing heat its pressure P and volume V are related by the equation PV 4 C where C is a constant Suppose that at a certain instant the volume is 300 cm and the pre is 80 kPa and is decreasing at a rate of 10 kPa min At what rate is the volume increasing at this instant Round your answer to the nearest whole number cm min Enter an integer or decimal number more Question Help Message instructor

find f x pant 10 1 y x 7X 2

A swimming pool is 20 ft wide 40 ft long 3 ft deep at the shallow end and 9 ft deep at its deepest point A cross section is shown in the figure If the pool is being filled at a rate of 1 ft min how fast the water level rising when the depth at the deepest point is 5 ft Round your answer to five decimal place ft min 6 12 16 6 Question Help Message instruct 3

d the equation of the tongent line using the point 7 ysin x X os 3x

X 1 Y tan X 1 X 1 Y 26

A neutral hollow spherical conducting shell of inner radius 1 00 cm and outer radius 3 00 cm has a 2 00 C point charge placed at its center a Find the surface charge density on the inner surface of the shell b Find the surface charge density on the outer surface of the shell

Find g 1 if g x x sin g x x and g 1 0

For each problem find f x by using the formula 7 f x x x 3 dy 1 dx 9 where y f x dx dy 8 f x 3x 2x 5

For each problem find x by using the theorem f x 1 x 3 f x 5x 1 5 f x 2x 3 4 f x 2x 2 6 f x 4x 4

y t an x

For each problem find 3 f x 5x 1 1 x f x by using the theorem x 4 f x 2x 2

For each problem find f x by direct computation 1 f x 3x 3 2 f x 2x

Find the derivative of the inverse of the following function at the specified point on the graph of the inverse function You do not need to find f 2 T 2 4 f x cos x The derivative of the inverse of the function f x cos x at Type an exact answer Points 0 of 1 2 T 24

Find the derivative of the inverse of the following function at the specified point on the graph of the inverse function You do not need to find f f x 4x 2 14 4 The derivative is

Given the function f find the slope of the line tangent to the graph of f at the specified point on the graph of f f x 5x The slope of the tangent line to the graph f is

Find the inverse of the following function and write it in the form f x 3 f x 2 Xx 1 for x 20 The inverse function is f x for 0 x 3

a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 6 a f x 0 5 9

6 The function f x is one to one 5 X a Find its inverse and check your answer b Find the domain and the range of f and f a f x 5 Simplify your answer 6 X b Find the domain of f Select the correct choice below and if necessary fill in the answer box to complete yo choice O A The domain is x x Type integers or fractions Use a comma to separate answers as needed OB The domain is x x Type integers or fractions Use a comma to separate answers as needed OC The domain is x x Type integers or fractions Use a comma to separate answers as needed D The domain is the set of all real numbers

1 Consider the functions f x 5x 5 and g x x 5 a Find f g x b Find g f x c Determine whether the functions f and g are inverses of each other Use a comma to separate answers as needed B No values should be excluded from the domain OA x b What is g f x g f x X Simplify your answer Give any values of x that need to be excluded from g f x Select the correct choice below and fill in any answer boxes within your choice Use a comma to separate answers as needed B No values should be excluded from the domain 1 2 of 3 c Are the functions f and g inverses of each other Choose the correct answer below Yes No Save for the centra Previous

sentence below If the domain of a one to one function f is 1 00 the range of its inverse f is 1 If the domain of a one to one function f is 1 00 the range of its inverse f is 1 0 00 m 11

For f x a fog 2 X 9 a fog x A and g x b gof 2X 5 9X X find the following composite functions and state the domain of each c fof d gog Simplify your answer Select the correct choice below and fill in any answer boxes within your choice The domain of fogis x x 0 x s x ww 5 1 Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B The domain off og is all real numbers 4 2 x 9 b gof x 2 9x 81 Simplify your answer Use integers or fractions for any numbers in the expression

If the shortest side of a 30 60 90 triangle has length 500 kilometers then what is the length of the longest side km

Use the following functions f x x 4 and g x x 2 to solve each of the following 5 f x 0 7 f x g x x 4 x 2 x 4 0 4 4 x 6 g x 0 X 2 0 2 a X 2

2 11 x 3x 10 0

A C E 1 y x 2 2 9x 4y z 1 B D F

Determine whether or not the sequence is geometric If it is find the common ratio r If an answer does not exist enter DNE 3 4 5 6 5 7 9 11 O Yes the sequence is geometric O No the sequence is not geometric r

Find a formula for an for the arithmetic sequence a 21 d 7 an

Rewrite cot 0 csc 20 csc 20 in terms of sine and cosine Simplify your answer

Evaluate the integral by interpreting it in terms of areas L 3 81 x dx

5 4 3 2 1 2 3 Select the name of the toolkit function in the graph Select an answer Select the equation of the toolkit function in the graph Of x 2 Of t 2 Of x 10 Of x log z Of x 2 Of x x Of x x Give the domain of the function in the graph Give the range of the function in the graph

Use Pascal s Triangle to find the binomial coefficient 5C2

Calculate the binomial coefficient 5

A sample of argon gas has a volume of 745 mL at a pres sure of 1 30 atm and a temperature of 132 C What is the final volume of the gas in milliliters when the pressure and temperature of the gas sample are changed to the following if the amount of gas does not change 0 55 atm and 81 C Express your answer using two significant figures V VAE