Evaluate the following integrals. ∫- π /4 π/4 sec2 x dx
> Differentiate (with respect to t or x): y = sin x tan x
> Differentiate (with respect to t or x): y = tan (sin x)
> Differentiate (with respect to t or x): y = sin (tan x)
> Differentiate (with respect to t or x): y = tan e-2x
> Differentiate (with respect to t or x): y = tan (x4 + x2)
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = √(x2 + y2)
> Differentiate (with respect to t or x): f (x) = tan3 2x
> Differentiate (with respect to t or x): f (x) = cos3 4x
> Differentiate (with respect to t or x): f (x) = (cos x – 1) / x3
> Differentiate (with respect to t or x): f (x) = cos 2x / sin 3x
> Differentiate (with respect to t or x): g(x) = sin(-2x) cos 5x
> Differentiate (with respect to t or x): g(x) = x3 sin x
> Differentiate (with respect to t or x): f (t) = cos t3
> Differentiate (with respect to t or x): f (t) = sin√t
> Differentiate (with respect to t or x): f (t) = sin 3t
> Differentiate (with respect to t or x): f (t) = 3 sin t
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = (x – y)/(x + y)
> Determining the height of a Tree A tree casts a 60-foot shadow when the angle of elevation of the sun (measured from the horizontal) is 53˚ . How tall is the tree?
> Geometry of a roof A gabled roof is to be built on a house that is 30 feet wide so that the roof rises at a pitch of 23˚ . Determine the length of the rafters needed to support the roof.
> When π /2 < t < π, is sin t positive or negative?
> When – π/2 < t < 0, is tan t positive or negative?
> Find the four values of t between -2π and 2π at which sin t = -cos t.
> Find the four values of t between -2π and 2π at which sin t = cos t.
> If cos t = - 2/3, what are the possible values for sin t?
> If sin t = 1/5, what are the possible values for cos t?
> The point with the given coordinates determines an angle of t radians, where 0 ≤ t ≤ 2π. Find sin t, cos t, and tan t. (3, -4)
> The point with the given coordinates determines an angle of t radians, where 0 ≤ t ≤ 2π. Find sin t, cos t, and tan t. (-.6, -.8)
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = ln(xy)
> Let f (x, y, z) = x2 e(y2+z2). Compute f (1,-1, 1) and f (2, 3,-4).
> The point with the given coordinates determines an angle of t radians, where 0 ≤ t ≤ 2π. Find sin t, cos t, and tan t. (-.6, .8)
> The point with the given coordinates determines an angle of t radians, where 0 ≤ t ≤ 2π. Find sin t, cos t, and tan t. (3, 4)
> Construct angles with the following radian measure. -9π/2
> Construct angles with the following radian measure. 5π/4
> Give the triangle interpretation of sin t, cos t, and tan t for t between 0 and π/2.
> Give the formula for converting degree measure to radian measure.
> Explain the radian measure of an angle.
> What are the derivatives of sin g(t), cos g(t), and tan g(t)?
> State an identity involving tan t and sec t.
> Define cot t, sec t, and csc t for an angle of measure t.
> Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = x ex2y2
> State as many identities involving the sine and cosine functions as you can recall.
> Give verbal descriptions of the graphs of sin t and cos t.
> What does it mean when we say that the sine and cosine functions are periodic with period 2π?
> Define sin t, cos t, and tan t for an angle of measure t for any t.
> Describe cot t for 0 < t < π/2 as a ratio of the lengths of the sides of a right triangle.
> If 0 < t < π/2, use Fig. 3 to describe sec t as a ratio of the lengths of the sides of a right triangle.
> Evaluate the following integrals. ∫3 / cos2 (2x) dx
> Evaluate the following integrals. ∫ 1 / cos2 x dx
> Evaluate the following integrals. ∫- π /8 π/8 sec2 (x + π/8) dx
> Physicians, particularly pediatricians, sometimes need to know the body surface area of a patient. For instance, they use surface area to adjust the results of certain tests of kidney performance. Tables are available that give the approximate body surfa
> 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(-2x) dx
