Application of derivatives Questions and Answers

Math 126 Homework 15 2 12 pts Find all zeros real and non real Completely factor the polynomial into a polynomial with linear factors q x 4x 20xXx 4x 13

A polynomial P is given P x x 81 a Find all zeros of P real and complex Enter your answers as a comma separated list Enter all answers including repetitions X b Factor P completely P x

Verify the identity sin 8 tan 8 Use a Reciprocal Identity to rewrite the expression in terms of sine and cosine and then simplif sin 8 sin 8 tan 8 cos 8 sin 8 cos 8

Simplify the trigonometric expression sec x 1 sec x 84 222

Prove the identity sin x sin x Use the Subtraction Formula for Sine and then simplify sin x 1 sin x sin x 1 cos x sin T Cos x 0

Intervals of Concavity and Points of Inflection Answer the following questions using the function f x 4x 2x 4 1 z f has points of inflection that occur at NOTE If you believe the answer is none type none Otherwise state the values as a comma separated list if needed f is concave down on f is concave up on NOTE State the open intonale

Verify the identity csc x sec x cos x cot x tan x cos x sin x

Make the indicated trigonometric substitution in the given algebraic expression and simplify see Example 7 Assume that 0 0 1 2 9 x x 3 sin 0

4 5 points Consider the following function f x 5x 7x 3 a 3 points Determine the equation of the tangent line to f x at x 1 b 1 point Use your result from part a to approximate f 1 5 c 1 point Calculate the second derivative of f x

Prove the cofunction identity using the Addition and Subtraction Formulas tan u cot u Since tan is undefined use a Reciprocal Identity and then use the Substitution Formulas to sim tan 7 u 11 sin COS EN cos u cos sin u cos cos u sin sin u sin u cos u o sin u 0 cos u 1 sin u

Use an Addition or Subtraction Formula to find the exact value of the expression as demonstrated In Example 1 cos 285

3 Determine the domain Express your answer in interval notation f x 81 1

1 Solve analytically Express your answer in interval notation S

Sketch a right triangle corresponding to the trigonometric function of the acute angle 0 Use the Pythagorean Theorem to de 16 3 cos 0 sin 0 0 1875 tan 0 sec 0 csc 0 cot 8 X X X X

he sonar of a navy cruiser detects a submarine that is r 4400 feet from the cruiser The angle between the water level and the submarine is 0 29 5 How deep is the submarine ft

Find the period and amplitude y 2 cos 3 COS period amplitude TTX 3

Use the angle conversion capabilities of a graphing utility to convert the angle measure to decimal degree form Round your answer to three decimal places if necessary 372 24 45 W O

Rewrite each angle in degree measure Do not use a calculator 7 a 2 b 4 15 O

While traveling across flat land you notice a mountain directly in front of you The angle of elevation to the peak is 1 5 After you drive 17 miles closer to the mountain the angle of elevation is 8 Approximate the height of the mountain ft

Compute Ay and dy for the given values of x and dx Ax Round your answers to three decimal places y ex x 0 Ax Ay dy 2 0 Sketch a diagram showing the line segments with lengths dx dy and Ay y 1 5 0 6 dx Ay y 2 0 K 1 5 dy Ay

Find the linearization L x of the function at a f x x 6x a 1 L x

Find the angle of depression from the top of a lighthouse 285 feet above water level to the water line of a ship 2 miles offshore

Find a and d for the function f x a cos x d such that the graph of f matches the figure X 11 d 4 X 6 47

Find the values of the six trigonometric functions of 0 with the given constraint If an answer is undefined enter UNDEFINED Function Value Constraint sin 8 0 SOS sin 80 cos 8 1 tan 0 0 CSC 8 sec 8 cot 8 x X x 31

Determine whether the statement is true or false g x g 2 X 2 If g x x5 then lim X 2 True False 80

Determine whether the statement is true or false d tan x sec x dx dx True False Submit Answer

Determine whether the statement is true or false If y e2 then y 2e True False

Determine whether the statement is true or false d If f and g are differentiable then dx True False f g x f g x g x

For the polynomial function f x 7x x 11 answer the following questions a List each real zero and its multiplicity b Determine whether the graph crosses or touches the x axis at each x intercept c Determine the maximum number of turning points on the graph d Determine the end behavior that is find the power function that the graph of f resembles for large values of x Juni VIIVIVU OA The real zero of f is with multiplicity Type an exact answer using radicals as needed Type integers or fractions OB The smallest zero of f is with multiplicity The largest zero of fis Type an exact answer using radicals as needed Type integers or fractions OC The smallest zero of f is with multiplicity The middle zero of f is largest zero of fis with multiplicity Type an exact answer using radicals as needed Type integers or fractions D There are no real zeros with multiplicity with multiplicity The

Construct a polynomial function that might have the given graph Choose the correct answer below OA f x x x 3 x 5 OB f x x x 3 x 5 OC f x x x 3 x 5 OD f x x x 3 x 5

For the polynomial function f x 6 x 7 x 6 2 answer the following questions a List each real zero and its multiplicity b Determine whether the graph crosses or touches the x axis at each x intercept c Determine the maximum number of turning points on the graph d Determine the end behavior that is find the power function that the graph of f resembles for large values of x a Find any real zeros of f Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The real zero of fis with multiplicity Type an exact answer using radicals as needed Type integers or fractions The largest zero of f is with multiplicity OB The smallest zero of f is with multiplicity Type an exact answer using radicals as needed Type integers or fractions with multiplicity The middle zero of f is OC The smallest zero of f is with multiplicity Type an exact answer using radicals as needed Type integers or fractions OD There are no real zeros The largest zero of f is with multiplicity

Construct a polynomial function f with the following characteristics zeros 1 1 and 4 degree 3 y intercept 12 Choose the correct answer below O A f x 3 x 1 x 1 x 4 O B f x x 1 x 1 x 4 O C f x 3 x 1 x 1 x 4 OD f x 3 x 1 x 1 x 4

4 Use a transformation of the graph of y x to graph the function 4 h x 2 x 1 3 Choose the correct graph below O A OB H hatt HOFF H OC Ay G OD

The circular blade on a saw has a diameter of 6 25 inches and rotates at 4600 revolutions per minute a Find the angular speed of the blade in radians per minute radians per minute b Find the linear speed of the saw teeth in inches per minute as they contact the wood being cut Round your answer to one decimal p inches per minute

Match the function with its graph f x x cos x 2 N M A mm 2 4 6 4 2 0 Describe the behavior of the function as x approaches zero 2r

Construct an appropriate triangle to complete the table 0 0s 90 050sn 2 Function e deg 0 rad Function Value sin

Find the period and the amplitude y 8 cos 2x period amplitude 00 Need Help Read 2 Watch

Find the first partial derivatives of the following function 2u F u v w 3v 7w

Question 3 5 5 pts Find the absolute maximum and absolute minimum values of each of the following functions on the given intervals a f x 2x 3 5 on 3 4 b f x 2x 3x on 0 2

termine where the given function is concave up and where it is concave dc 7 f x x 9 x2 A Concave up on 3 0 concave down on 0 3 B Concave up on 0 concave down on 0 C Concave down on 3 0 concave up on 0 3 D Concave up on

d the absolute extreme values of the function on the interval 3 3 f x CSC X 75x5 37 2 A absolute maximum does not exist absolute minimum does not exist B absolute maximum is 0 at x 7 absolute minimum is 1 at x C absolute maximum is 1 at x t absolute minimum is 1 at x D absolute maximum is 1 at x t absolute minimum is 1 at x 0

4 Suppose a company has fixed costs of 60 000 and variable cost per unit of x 333 dollars where x is the total number of units produced Suppose further that the selling price 2363 dollars per unit a Find the break even points Enter your answers as a comma separated list X b Find the maximum revenue Round your answer to the nearest cent c Form the profit function P x from the cost and revenue functions P x Find maximum profit d What price will maximize the profit Round your answer to the nearest cent its product i

Determine all critical points for the function 1 f x x3 3x2 5 A x 0 2 y 2x2 64 x A x 0 and x 4 C x 0 B x 0 and x 1 C x 1 and x 1 B x 4 D x 0 x 4 and x 4 D x 0 and x 2

3 Find an equation of the tangent line to the curve In a at the point 1 0

Suppose that the only information we have about a function f is that f 1 7 and the graph of its derivative is as shown fo 2 2 3 x a Use a linear approximation to estimate f 0 9 and f 1 1 f 0 9 f 1 1 b Are your estimates in part a too large or too small Explain The slopes of the tangent lines are positive but the tangents are becoming less steep So the tangent lines lie above the curve f Thus the estimates are too small The slopes of the tangent lines are negative but the tangents are becoming steeper So the tangent lines lie below the curve f Thus the estimates are too small The slopes of the tangent lines are negative but the tangents are becoming steeper So the tangent lines lie below the curve f Thus the estimates are too large The slopes of the tangent lines are positive but the tangents are becoming less steep So the tangent lines lie above the curve f Thus the estimates are too large

2 A ball is thrown vertically upward with a velocity of 80 ft s then its height after t seconds is s 80t 16t a What is the maximum height reached by the ball b What is the velocity of the ball when it is 96 ft above the ground on its way up On its way down

Compute Ay and dy for the given values of x and dx Ax y x 6x x 5 Ax 0 5 Ay dy Sketch a diagram showing the line segments with lengths dx dy and Av

The circumference of a sphere was measured to be 74 cm with a possible error of 0 5 cm a Use differentials to estimate the maximum error in cm in the calculated surface area Round your answer to the nearest integer cm What is the relative error Round your answer to three decimal places b Use differentials to estimate the maximum error in cm in the calculated volume Round your answer to the nearest integer cm 3 What is the relative error Round your answer to three decimal places

a Find the differential dy y 15 x dy b Evaluate dy for the given values of x and dx x 1 dx 0 2 dy

The radius of a circular disk is given as 28 cm with a maximum error in measurement of 0 2 cm a Use differentials to estimate the maximum error in cm in the calculated area of the disk Round your answer to tw decimal places cm b What is the relative error Round your answer to four decimal places What is the percentage error Round your answer to two decimal places