The metal bar of length l in Figure 3.27 has one end attached at the point P to a circle

Construct a table of values for cos x, x = 0, 0.1, 0.2,..., 0.6. Using the difference quotient, estimate the derivative at these points (use h = 0.001), and compare it with sin x.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

In Exercises 247, find the derivatives of the functions. Assume a, b, and c are constants.

Is the graph of y = sin(x4) increasing or decreasing when x = 10? Is it concave up or concave down?

Find the line tangent to f(t) = 3 sin(2t)+5 at the point where t =

Find the 50th derivative of y = cos x

Find d2x/dt2 as a function of x if dx/dt = x sin x

Find a possible formula for the function q(x) such that q (x) = ex sin x ex cos x (sin x

Find a function F(x) satisfying F (x) = sin(4x)

Let f(x) = sin2 x + cos2 x. (a) Find f (x) using the formula for f(x) and derivative

Let f (x) = sin x2 , Find h(x) if h(x) = f x2 .

On page 39 the depth, y, in feet, of water in Boston harbor is given in terms of t, the number of hours since midnight, by y =5+4.9 cos  6 t  . (a) Find dy/dt. What does dy/dt represent, in terms of water level? (b) For 0 t 24, when is dy/dt zero? (Figure 1.55 on page 39 may be helpful.) Explain what it means (in terms of water level) for dy/dt to be zero.

A boat at anchor is bobbing up and down in the sea. The vertical distance, y, in feet, between the sea floor and the boat is given as a function of time, t, in minutes, by y = 15 + sin(2t). (a) Find the vertical velocity, v, of the boat at time t. (b) Make rough sketches of y and v against t.

The voltage, V , in volts, in an electrical outlet is given as a function of time, t, in seconds, by the function V = 156 cos(120t). (a) Give an expression for the rate of change of voltage with respect to time. (b) Is the rate of change ever zero? Explain. (c) What is the maximum value of the rate of change?

An oscillating mass at the end of a spring is at a distance y from its equilibrium position given by y = A sin k m  t  . The constant k measures the stiffness of the spring. (a) Find a time at which the mass is farthest from its equilibrium position. Find a time at which the mass is moving fastest. Find a time at which the mass is accelerating fastest. (b) What is the period, T , of the oscillation? (c) Find dT /dm. What does the sign of dT /dm tell you?

With t in years, the population of a herd of deer is represented by P(t) = 4000 + 500 sin  2t 2  . (a) How does this population vary with time? Graph P(t) for one year. (b) When in the year the population is a maximum? What is that maximum? Is there a minimum? If so, when? (c) When is the population growing fastest? When is it decreasing fastest? (d) How fast is the population changing on July 1?

An environmentalist reports that the depth of the water in a new reservoir is approximated by h = d(t) = kt 0 t 2 50 + sin(0.1t) t > 2, where t is in weeks since the date the reservoir was completed and h is in meters. (a) During what period was the reservoir filling at a constant rate? What was that rate? (b) In this model, is the rate at which the water level is changing defined for all times t > 0? Explain.

The metal bar of length l in Figure 3.27 has one end attached at the point P to a circle of radius a. Point Q at the other end can slide back and forth along the x-axis. (a) Find x as a function of . (b) Assume lengths are in centimeters and the angular speed (d/dt) is 2 radians/second counterclockwise. Find the speed at which the point Q is moving when (i) = /2, (ii) = /4.

Find the equations of the tangent lines to the graph of f(x) = sin x at x = 0 and at x = /3. Use each tangent line to approximate sin(/6). Would you expect these results to be equally accurate, since they are taken equally far away from x = /6 but on opposite sides? If the accuracy is different, can you account for the difference?

If k 1, the graphs of y = sin x and y = kex intersect for x 0. Find the smallest value of k for which the graphs are tangent. What are the coordinates of the point of tangency?

A wave travels along a string that is joined to a thicker rope. The wave both reflects back along the string and is transmitted to the rope. For positive constants k1, k2, w and time t, the wave along the string, given by x < 0, is f(x) = cos(k1x wt) + R cos(k1x wt) and the wave along the rope, given by x > 0, is g(x) = T cos(k2x wt). For every value of t, the two waves have the same tangent line at x = 0, so they have the same value and same slope at x = 0. Use this fact to show that R = k1 k2 k1 + k2 and T = 2k1 k1 + k2 . These amplitudes are called the reflection coefficient, R, and the transmission coefficient, T

We will use the following identities to calculate the derivatives of sin x and cos x: sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b sin a sin b. 3.5 THE TRIGONOMETRIC FUNCTIONS 155 (a) Use the definition of the derivative to show that if f(x) = sin x, f (x) = sin x lim h0 cos h 1 h + cos x lim h0 sin h h . (b) Estimate the limits in part (a) with your calculator to explain why f (x) = cos x. (c) If g(x) = cos x, use the definition of the derivative to show that g (x) = sin x.

In this problem you will calculate the derivative of tan rigorously (and without using the derivatives of sin or cos ). You will then use your result for tan to calculate the derivatives of sin and cos . Figure 3.28 shows tan and (tan ), which is the change in tan , namely tan( + ) tan . (a) By paying particular attention to how the two figures relate and using the fact that Area of Sector OAQ Area of Triangle OQR Area of Sector OBR explain why 2 (cos )2 (tan ) 2 2 (cos( + ))2 . [Hint: A sector of a circle with angle at the center has area /(2) times the area of the whole circle.] (b) Use part (a) to show as 0 that tan  1 cos 2 , and hence that d(tan ) d =  1 cos 2 . (c) Derive the identity (tan ) 2 +1=  1 cos 2 . Then differentiate both sides of this identity with respect to , using the chain rule and the result of part (b) to show that d d (cos ) = sin .

In Problems 6869, explain what is wrong with the statement

In Problems 6869, explain what is wrong with the statement

In Problems 7071, give an example of

In Problems 7071, give an example of

Are the statements in Problems 7274 true or false? Give an explanation for your answer.

Are the statements in Problems 7274 true or false? Give an explanation for your answer.

Are the statements in Problems 7274 true or false? Give an explanation for your answer.