Find the following indefinite integrals. ∫sin(-2x) dx
> Evaluate the following integrals. ∫sec2 (2x + 1) dx
> Evaluate the following integrals. ∫sec2 3x dx
> Repeat Exercise 33(a) and (b) using the point (0, 0) on the graph of y = tan x instead of the point (Ï€/2, 1). Exercise 33: (a) Find the equation of the tangent line to the graph of y = tan x at the point (Ï€/4, 1). (b) Copy the port
> (a) Find the equation of the tangent line to the graph of y = tan x at the point (Ï€/4, 1). (b) Copy the portion of the graph of y = tan x for - Ï€/2 Figure 5: င်။ ၊ k 3 2 .,
> Differentiate (with respect to t or x): y = ln(tan t)
> Differentiate (with respect to t or x): y = ln(tan t + sec t)
> Differentiate (with respect to t or x): y = tan4 3t
> Differentiate (with respect to t or x): y = (1 + tan 2t)3
> Differentiate (with respect to t or x): y = √ tan x
> Differentiate (with respect to t or x): y = tan2 x
> Let f (x, y) = 3x2 + 2xy + 5y, as in Example 5. Show that f (1 + h, 4) - f (1, 4) = 14h + 3h2. Thus, the error in approximating f (1 + h, 4) - f (1, 4) by 14h is 3h2. (If h = .01, for instance, the error is only .0003.)
> Differentiate (with respect to t or x): y = e3x tan 2x
> Differentiate (with respect to t or x): y = x tan x
> Differentiate (with respect to t or x): y = 2 tan √(x2 – 4)
> Differentiate (with respect to t or x): y = tan√x y = 2 tan √(x2 – 4) y = x tan x y = e3x tan 2x y = tan2 x y = √ tan x y = (1 + tan 2t)3 y = tan4 3t y = ln(tan t + sec t) y = ln(tan t)
> Differentiate (with respect to t or x): f (x) = 3 tan (1 - x2)
> Differentiate (with respect to t or x): f (x) = 4 tan (x2 + x + 3)
> Differentiate (with respect to t or x): f (x) = 5 tan (2x + 1)
> Differentiate (with respect to t or x): f (x) = 3 tan (π - x)
> Differentiate (with respect to t or x): f (t) = tan πt
> Differentiate (with respect to t or x): f (t) = tan 4t
> Compute ∂2f/∂y2, where f (x, y) = 60 x3/4 y1/4, a production function (where y is units of capital). Explain why ∂2f/∂y2 is always negative.
> Differentiate (with respect to t or x): f (t) = cot 3t
> Differentiate (with respect to t or x): f (t) = cot t
> Differentiate (with respect to t or x): f (t) = csc t
> Differentiate (with respect to t or x): f (t) = sec t
> The angle of elevation from an observer to the top of a church is .3 radian, while the angle of elevation from the observer to the top of the church spire is .4 radian. If the observer is 70 meters from the church, how tall is the spire on top of the chu
> Find the width of a river at points A and B if the angle BAC is 90, the angle ACB is 40, and the distance from A to C is 75 feet. See Fig. 6. Figure 6:
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. (.8,-.6)
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. (-.6, -.8)
> Give the values of tan t and sec t, where t is the radian measure of the angle shown.
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. (-2, 2)
> Compute ∂2f/∂x2, where f (x, y) = 60 x3/4 y1/4, a production function (where x is units of labor). Explain why ∂2f/∂x2 is always negative.
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. (2, -3)
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. (-2, 1)
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. 3
> Give the values of tan t and sec t, where t is the radian measure of the angle shown. 13 5
> Differentiate (with respect to t or x): y = sin 4t
> The number of hours of daylight per day in Washington, D.C., t weeks after the beginning of the year is The graph of this function is sketched in Fig. 13. (a) How many hours of daylight are there after 42 weeks? (b) After 32 weeks, how fast is the numb
> The average weekly temperature in Washington, D.C., t weeks after the beginning of the year is The graph of this function is sketched in Fig. 12. (a) What is the average weekly temperature at week 18? (b) At week 20, how fast is the temperature changin
> As h approaches 0, what value is approached by cos( + h) +1, h
> As h approaches 0, what value is approached by sin (+h)-1 h -?
> The basal metabolism (BM) of an organism over a certain time period may be described as the total amount of heat in kilocalories (kcal) that the organism produces during this period, assuming that the organism is at rest and not subject to stress. The ba
> For the production function f (x, y) = 60 x3/4 y1/4 considered in Example 8, think of f (x, y) as the revenue when x units of labor and y units of capital are used. Under actual operating conditions, say, x = a and y = b, ∂f/∂x (a, b) is referred to a
> A person’s blood pressure P at time t (in seconds) is given by P = 100 + 20 cos 6t. (a) Find the maximum value of P (called the systolic pressure) and the minimum value of P (called the diastolic pressure) and give one or two values of t where these maxi
> Find the area under the curve y = sin 2t from t = 0 to t = π/4.
> Find the area under the curve y = cos t from t = 0 to t = π/2.
> Find the following indefinite integrals. ∫ cos (x – 2)/2 dx
> Find the following indefinite integrals. ∫sin (4x + 1) dx
> Find the following indefinite integrals. ∫ (- sin x + 3 cos(-3x)) dx
> Find the following indefinite integrals. ∫ (2 sin 3x + cos 2x /2) dx
> Find the following indefinite integrals. ∫ (cos x - sin x) dx
> Find the following indefinite integrals. ∫2 sin x/2 dx
> Richard Stone (see Exercise 32) determined that the yearly consumption of food in the United States was given by f (m, p, r) = (2.186) m0.595 p-0.543 r0.922. Determine which partial derivatives are positive and which are negative, and give interpretati
> Find the following indefinite integrals. ∫-1/2 cos x/7 dx
> Find the following indefinite integrals. ∫3 sin 3x dx
> Find the following indefinite integrals. ∫cos 2x dx
> Find the equation of the line tangent to the graph of y = 3 sin 2x - cos 2x at x = 3 π/4.
> Find the equation of the line tangent to the graph of y = 3 sin x + cos 2x at x = π/2.
> Find the slope of the line tangent to the graph of y = sin 2x at x = 5 π/4.
> Find the slope of the line tangent to the graph of y = cos 3x at x = 13π/6.
> Differentiate (with respect to t or x): y = (cos t) ln t
> Differentiate (with respect to t or x): y = sin(ln t)
> Differentiate (with respect to t or x): y = ln(sin 2t)
> Using data collected from 1929 to 1941, Richard Stone determined that the yearly quantity Q of beer consumed in the United Kingdom was approximately given by the formula Q = f (m, p, r, s), where f (m, p, r, s) = (1.058) m0.136 p-0.727 r0.914 s0.816 a
> Differentiate (with respect to t or x): y = ln(cos t)
> Differentiate (with respect to t or x): y = cos (e2x+3)
> Differentiate (with respect to t or x): y = sin t / cos t
> Differentiate (with respect to t or x): y = (1 + x) /cos x
> Differentiate (with respect to t or x): y = sin 2x cos 3x
> Differentiate (with respect to t or x): y = (cos x + sin x)2
> Differentiate (with respect to t or x): y = ex sin x
> Differentiate (with respect to t or x): y = cos2 x + sin2 x
> Differentiate (with respect to t or x): y = cos2 x3
> Differentiate (with respect to t or x): y = 3√(sin π t)
> The volume (V) of a certain amount of a gas is determined by the temperature (T) and the pressure (P) by the formula V = .08(T/P). Calculate and interpret ∂V/∂P and ∂V/∂T when P = 20, T = 300.
> Differentiate (with respect to t or x): y = (1 + cos t)8
> Differentiate (with respect to t or x): y = ecos x
> Differentiate (with respect to t or x): y = √ sin(x - 1)
> Differentiate (with respect to t or x): y = cos(ex)
> Differentiate (with respect to t or x): y = sin √(x – 1)
> Differentiate (with respect to t or x): y = sin3 t2
> Differentiate (with respect to t or x): y = cos3 t
> Differentiate (with respect to t or x): y = cos (2x + 2) / 2
> Differentiate (with respect to t or x): y = sin (π - t)
> Differentiate (with respect to t or x): y = t cos t
> The demand for a certain gas-guzzling car is given by f (p1, p2), where p1 is the price of the car and p2 is the price of gasoline. Explain why ∂f/∂p1 < 0 and ∂f/∂p2 > 0.
> Let g (x, y, z) = x/(y - z). Compute g (2, 3, 4) and g (7, 46, 44).
> Differentiate (with respect to t or x): y = t cos t
> Differentiate (with respect to t or x): y = - sin 3t / 3
> Differentiate (with respect to t or x): y = 2 cos 3t
> Differentiate (with respect to t or x): y = cos (- 4t)
> Differentiate (with respect to t or x): y = 4 sin t
> Differentiate (with respect to t or x): y = 2 cos 2t
> Give the values of sin t and cos t, where t is the radian measure of the angle shown. 16 4 1
> Give the values of sin t and cos t, where t is the radian measure of the angle shown. 3 2
> Give the values of sin t and cos t, where t is the radian measure of the angle shown. t √5 2
> Give the values of sin t and cos t, where t is the radian measure of the angle shown. 2 t √3 1
> Let p1 be the average price of MP3 players, p2 the average price of audio files, f (p1, p2) the demand for MP3 players, and g (p1, p2) the demand for audio files. Explain why ∂f/∂p2 < 0 and ∂g/∂p1 < 0.
> In any given locality, the length of daylight varies during the year. In Des Moines, Iowa, the number of minutes of daylight in a day t days after the beginning of a year is given approximately by the formula (Source: School Science and Mathematics.) (
