Calculus Questions

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

Prove the identity sin x 7 cos x 2 Use the Subtraction Formula for Sine and then simplify sin x sin x cos cos x CO sin x 0 cos x

Find the area between the curves y ex and y e x shown below from x 0 to x 1 Leave the answer in its exact form y 21 18 15 12 9 6 3 0 5 y e x y e

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 sin w sin w cos w tan w 1 tan w

Verify the identity sin t cos t 2 sec t csc t sin t cos t sin t cos t sin t cos t sin t 2 sin t cos t sin t cos t sin t cos t sin t cos t 1 sin t cos t 2 sec t cs t sin t cos t

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

Sketch a graph of f z 0 5z 21 1 Before sketching the graph determine where the function has its minimum or maximum value so you can place your first point there 5 4 3 2 1 5 4 3 2 1 Clear All Draw 1 2 3 4 5 2 The zeros of the function are at the values The x intercept s are at the points The y intercept is at the point

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

77 x Analyze the polynomial function f x The graph of f behaves like y 32 7x 7x 21 Answer parts a through h www for large values of x b Find the x and y intercepts of the graph of the function The x intercept s is are 7 3 7 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once the x axis at x The y intercept s is are 21 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once c Determine the zeros of the function and their multiplicity Use this information to determine whether the graph crosses or touches the x axis at each x intercept The least zero is of multiplicity so the graph of f so the graph of f the x axis at x The real zero s of f is are 7 3 7 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once Sa the x axis at x The greatest zero is of multiplicity so the graph of f The middle zero is of multiplicity

Concavity and points of inflection Answer the following questions using the function f x e The first derivative of the given function is The second derivative of the given function is x 0 f has points of inflection that occur at x NOTE If you believe the answer is none type none Otherwise state the values as a comma separated list if needec f is concave down on ar and concave up on NOTE State the open intervals of concavity as a comma separated list if needed

1 3 3 Analyze the polynomial function f x 7x7x 7x 21 Answer parts a through h a Determine the end behavior of the graph of the function 1 3 X for large values of x The graph of f behaves like y www b Find the x and y intercepts of the graph of the function The x intercept s is are Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answe only once h

4 Find an antiderivative of the given function a f x x x b f x 3e2x x 1 x d f x 2 cos x e f x sin 3x x

Bonus 5 points Write a function such that f x f x 0 e

2 1 point Write 3 different equations f x g x and h x such that 2 f x g x h x x

5 3 points 1 5 points each Determine the derivative of each function a f t 24 e 7 b g x e 5 67

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 identity tan x 1 3 x T 3 1 3 tan x 1 3 tan x tan x tan 3 3 tan x 1 3 tan x

Prove the identity sin x sin x 2

dw dt t Use the Chain Rule to find where w cos 8x sin 2y x and y t 4 aw ax Type an expression using x and y as the variables

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

Prove the identity sin x 7 sin x Use the Subtraction Formula for Sine and then simplify sin x sin x sin x It II cos x sin 77 1 cos x 0

Prove the identity tan x 7 tan x Use the Subtraction Formula for Tangent and then simplify tan x tan tan x Tt tan x 1 1 tan x 0 tan x

Prove the identity sin x cos x 2 Use the Subtraction Formula for Sine and then simplify sin x 2 sin x cos cos x sin x 0 cos x

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

2 Solve analytically Express your answer in interval notation 3x 4 1 8

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

Complete the identity csc 0 1

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

The radius of a circular disk is given as 29 cm with a maximum error in measurement of 0 2 cm a Use differentials to estimate the maximum error in the calculated area of the disk Round your answer to two 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

Match the function with its graph f x x cos x 4 2 4 2 Mw 2T 2 0 2 6 4 2 0 5 2 And 2 0 2 2 6 10 6 Describe the behavior of the function as x approaches zero As x 0 f x 2T 2 6 6 2 0 5 5 2 S 2T 2T

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

Solve the right triangle shown in the figure A B C a 9 C 69 16 B C O O b C

a Find the differential dy y 3 x dy b Evaluate dy for the given values of x and dx x 1 dx 0 1 dy

For the polynomial function f x 6 x 7 x 6 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 B The smallest zero of f is 6 with multiplicity 2 The largest zero of f is 7 with multiplicity 1 Type an exact answer using radicals as needed Type integers or fractions OC The smallest zero of fis with multiplicity The middle zero of f is Type an exact answer using radicals as needed Type integers or fractions with multiplicity The largest zero of f is D There are no real zeros b Select the correct choice below and if necessary fill in the answer box es to complete your choice Points 0 5 of 1 c The maximum number of turning points on the graph is Type a whole number with multiplicity OA The graph crosses the x axis at Type an exact answer using radicals as needed Type an integer or a simplified fraction Use a comma to separate answers as needed B The graph touches the x axis at Type an exact answer using radicals as needed Type an integer or a simplified fraction Use a comma to separate answers as needed C The graph touches the x axis at 6 and crosses at 7 Type an exact answer using radicals as needed Type integers or simplified fractions Use a comma to separate answers as needed OD The graph neither crosses nor touches the x axis

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 d dx True False In 10 1 10

Determine whether the statement is true or false 10x X10X 1 d dx True False

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