Find ∂f/∂x and ∂f/∂y for each of the following functions. f (x, y) = (x – y)/(x + y)
> Evaluate the following integrals. ∫-π π (cos 3x + 2 sin 7x) dx
> Evaluate the following integrals. ∫0 π (x - 2 cos (π - 2x))
> Evaluate the following integrals. ∫ cos (6 - 2x) dx
> Evaluate the following integrals. ∫0 π /2 cos 6x dx
> Evaluate the following integrals. ∫ (3 cos 3x - 2 sin 2x) dx
> Let f (x, y, z) = (1 + x2y)/z. Find ∂f/∂x, ∂f/∂y, and ∂f/∂z.
> Evaluate the following integrals. ∫ sin (π - x) dx
> A spirogram is a device that records on a graph the volume of air in a person’s lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace
> A spirogram is a device that records on a graph the volume of air in a person’s lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace
> A spirogram is a device that records on a graph the volume of air in a person’s lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace
> Find the area of the region bounded by the curves y = x and y = sin x from x = 0 to x = π.
> Find the area of the region between the curve y = cos t and the t-axis from t = 0 to t = 3π/2.
> Find the area of the region between the curve y = sin t and the t-axis from t = 0 to t = 2π.
> Find the area under the curve y = 2 + sin 3t from t = 0 to t = π/2.
> Sketch the graph of y = t + sin t for 0 ≤ t ≤ 2π.
> Find the equation of the line tangent to the graph of y = tan t at t = π/4.
> Let f (p, q) = 1 - p(1 + q). Find ∂f/∂q and ∂f/∂p.
> The identity sin (s + t) = sin s cos t + cos s sin t was given in Section 8.2. Compute the partial derivative of each side with respect to t, and obtain an identity involving cos (s + t).
> If f (s, t) = t sin st, find ∂f/∂s and ∂f/∂t.
> If z = sin wt, find ∂z/∂w and ∂z/∂t.
> If f (s, t) = sin s cos 2t, find ∂f/∂s and ∂f/∂t.
> Show that y = 3 sin 2t + cos 2t satisfies the differential equation y’’ = -4y.
> If f (t) = sin2 t, find f ‘(t).
> f (t) = et tan t
> f (t) = etan t
> Differentiate (with respect to t or x): f (t) = tan 2t / cos t
> Differentiate (with respect to t or x): f (t) = sin t / tan 3t
> Let f (L, K) = 3√(LK). Find ∂f/∂L.
> Differentiate (with respect to t or x): y = sin4 e3x
> Differentiate (with respect to t or x): y = e3x sin4 x
> Differentiate (with respect to t or x): y = ln (cos x)
> Differentiate (with respect to t or x): y = ln (sin x)
> Differentiate (with respect to t or x): y = ln x cos x
> 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
> 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
> Evaluate the following integrals. ∫- π /4 π/4 sec2 x 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)
