Differentiation Questions and Answers

The point P 1 3 lies on the function of f x 4x x If Q is the point x 4x x graph the function f x and the secant lines PQ passing through P 1 3 for each value of x below Also find the slope for each value Only a simplified fraction will be accepted If x 2 the slope of PQ is If x 1 5 the slope of PQ is If x 0 5 the slope of PQ is Graph 9 8 7 6 5 4 3 2

interpolation coefficients required to calculate the derivative at 0 7 X 0 1 0 4 0 7 1 1 3 1 6 1 9 2 2 f x Sy 1 2 y 3 6 7 9 13 14 22 24 9 1 O 7 1 1 1 8 2 4 1 8 2 1 5 115 8 8 1 2 3 7 O 1712 1 2 1 5 15 38 O2 1 2 1 5 15 38 6 8 y 1 2 S y 9 1 5 10 13 8 6 15 23 85y 1 2 2 21 38

A woman is walking along a sidewalk cleaning up litter and once her bag is full she walks back to where she started On the way out while gathering litter she moves at 8 feet per minute On the way back she walks much faster at 80 feet per minute If the whole trip took 1 hour how far did she go from her starting point

8 points Find a polynomial function h x of least degree 3 having only real coefficients and zeros as given Assume multiplicity 1 and leading coefficient 1 Don t leave the answer in factor form Zeros of 1 and 4 2i

54 8 928 2 5 669 6 Question 17 Five Steel Parts weigh 2 5 lbs How many of these parts are in a box weigh lbs 69 84 68 50

Now TT 3 2 0 sin 2017 2 sin sin 1 4 After simplifying we have SEP 2 10 dt 2 T

We have 1201031 23 1 1 cos 20 de sin 4

Part 5 of Since we have an even power of cos x to evaluate cos 0 Submit Skip you cannot come back cos 3 cos2 e de we must now make the further substitution 0 J 4

Using t sec 0 we get 2 1 1 2 P 1 J 4 This can be further simplified to T 3 J 4 or simply T 3 A J 4 dt sec 8 cos 0 de TT 3 de sec 0 tan 0 sec 0 tan O de

1 Determine the corresponding squared conversion statements for each conversion a 1760 yards 1 mile b 5280 feet 1 mile c 1 yard 0 9144 m c 2 Determine the corresponding cubed conversion statements for each conversion a 1 ft 30 48 cm d 1000 m 1 km b 1 mile 1 609 km I d 100 mm 1 cm 12 inch 1 foot

Evaluate the integral using integration by parts with the indicated choices of u and dv 1 2x In xdx u ln x dv 2x dx C

dz Consider the equation 4 sin x y 5 sin x z 5 sin y z 0 Find the values of and ox dz Ox 2x 21 2x Simplify your answer Type an exact answer using radicals as needed dz dy at the point 2 2 2

dw 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 dt Then b evaluate dw dt at the given value of t 2 2 w 4x 10y x cost y sint t 3 dw a Express as a function of t dw dt

For the function f x y af 9 D1 7 2 x 6y of find and dy

20 UN Security Council never votes to use military power against another memb True or False Reading UN Funds Programs and Specialized Agencies False

Find all solutions of the equation in the interval 0 2 cos 0 0 4857 If there is more than one solution separate them with commas Do not round any intermediate computations Give your answer s in radians and round your answer s to the nearest hundredth 0 0 0 0

Rewrite cos tan w as an algebraic expression in w cos tan w S

Use the Second Fundamental Theorem of Calculus to find F x X 1 E 3 dt 4 7 F x F x

Evaluate the definite integral 649 e 36 Submit Answer dx xv In x

Evaluate the definite integral x 1 2x 4 dx

Evaluate the indefinite integral e7x 12042 e7x 2 dx C

Now if u 2 3x then dx xx du 1 du 2 3x This evaluates as the following Remember to use absolute values where appropriate 11 du

Step 1 We must decide what to choose for u If u f x then du f x dx and so it is helpful to look for some expression in missing a constant factor We see that 2 3x is part of this integral and the derivative of 2 3x is dx for which the derivative is also present though perhaps 3x dx 20 x 3x 2 2 3x which is simply a constant

Evaluate the integral by making the given substitution x 8 x 6 dx u 8 x x4 6 Part 1 of 4 We know that if u f x then du f x dx Therefore if u 8 x4 then du Orchmit Skin you coppet come back dx

An alternating series always converges True False

Two cards are drawn with replacement from a deck of cards Find the probability of selecting a king and a face card in order 0 083 O 0 143 0 018 0 374 0 247

A student takes a multiple choice test of 10 questions with four choices and guesses on each question Find the probability that the student gets exactly 7 correct answers 0 183 O 0 047 O 0 029 0 003 0 015

There are 8 women and 6 men in a department A committee of six people is to be selected How many different ways can the committee be selected if there must be 3 women and 3 men 12 720 3850 076 O2780

A shirt company has 5 designs each of which can be made with short or long sleeves There are 8 color patterns available How many different types of shirts are available from this company Write the answer as a whole number

Find the future value of the loan Round the answer to the nearest dollar 9570 at 4 35 for 8 years

Find the absolute minimum and absolute maximum values of f on the given interval 281 81 81 Step 1 The absolute maximum and minimum values occur either at a critical point inside the interval or at an endpoint of the interval Recall that a critical point a point where f x 0 We begin by finding the critical points x2 81 2x x 81 x 81 f x f x Step 2 After simplifying we have f x x 81 x 81 7 2x

F x x4 5 x 9 X Enter your answers as a comma separated list If an answer does not exist enter DNE Enhanced Feedback To find critical numbers of a differentiable function you need to differentiate the given function F x and solve for x in the equation F x 0 All solutions as we not differentiable are the critical numbers Make sure that you use the Product Rule when differentiating the function

x 1 Step 1 Recall the Inverse Substitution Rule where f and g are differentiable functions and g is one to one R x dx We are given the following dx 1 2 x x37 2 If f g t g t dt We note that x 1 2 x 1 Therefore the following entry from the Table of Trigonometric Substitutions is appropriate Expression It x a x a sec 8 0 8 or 0 3 2 2 1 2 a Substitution then a Identity sec 0 1 tan 0 Therefore we can let x sec 0 so dx de We also must make a substitution for the limits of integration in the definite integral Since x sec 8 we note that when x 2 0 Further when x 4 8 sec

ind f x for the given function f x 1 4 x

Find f x for the given function f x 7x

Use logarithmic differentiation to find the derivative of the function y x6 sin x y x

Sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f Enter your answers as a comma separated list If an answer does not exist enter f x 2 3x x 1 absolute maximum value absolute minimum value local maximum value s local minimum value s

y x y x6x X Enhanced Feedback y Third multiply y Please try again First take the logarithm on both sides to get an expression for In y Second differentiate both sides with respect to x to obtain an expression for the expression of y to finally get the expression for y In most cases when logarithmic differentiation

A cylindrical tank with radius 3 m is being filled with water at a rate of 2 m min How fast is the height of the water increasing in m min m min

Suppose f and g are differentiable functions and let h x f x 9 x then h x is given by the following form h x g x f x 91 f x In f x f x 9 g x Use this formula to find the derivative for each of the following functions a h x x5 h x b h x 5x h x c h x cos x h x

Let V represent the volume of a sphere with radius r mm Write an equation for V in mm in terms of r V r mm Find the radius of a sphere in mm when its diameter is 60 mm r mm The radius of a sphere is increasing at a rate of 5 mm s How fast is the volume increasing in mm s when the diameter is 60 mm Round your answer to two decimal p mm s

The radius of a sphere is increasing at a rate of 4 mm s How fast is the volume increasing in mm s when the diameter is 60 mm Round your answer to two decimal places m s mm

a If A is the area of a circle with radius r and the circle expands as time passes find dA dt dA dt dr dt in terms of dr dt b Suppose oil spills from a ruptured tanker and spreads in a circular pattern If the radius of the oil spill increases at a constant rate of 2 m s exactly how fast in m s is the area of th increasing when the radius is 29 m m s

Each side of a square is increasing at a rate of 7 cm s At what rate in cm s is the area of the square increasing when the area of the square is 9 cm cm s

Find the derivative of the function f x sin 8x f x X

Find the derivative of the function y tan x 5 y

Differentiate the function f x f x sin 7 In x

Question A sample of bacteria is being eradicated by an experimental procedure The population is following a pattern of exponential decay and approaching a population of O If the sample begins with 500 bacteria and after 11 minutes there are 200 bacteria after how many minutes will there be 50 bacteria remaining Round your answer to the nearest whole number and do not include units

The total amount paid on a 35 year loan was 98 000 If the interest rate was 4 1 and compounded monthly what w the principal Round your answer to the nearest dollar

2 Simplify the following 2sin2x sinx 1 sin2x sinx