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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Explain how the diagram in the figure illustrates integration by parts for definite integrals. V s = g(b)- r = g(a). 0 A B /u= u = f(v) v = g(u) U p = f(a) q = f(b) Area of A + Area of B = qs - pr
The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. Ja dx (ex + e-x)² 2
Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. X dx X и : n=x:
Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. √ dx V1 + √x ; x = (u² − 1)²
Evaluate the following integrals. Isi sin² x cos³ x dx
More composite functions Let f (x) = x, g(x) = x2 - 4, F(x) = √x, and G(x) = 1/(x - 2). Determine the following composite functions and give their domains.f ° G
Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing.|x| + |y| = 1
Determine whether the graphs of the following equations and functions have symmetry about the x-axis, the y-axis, or the origin. Check your work by graphing.f(x) = x|x|
More composite functions Let f (x) = x, g(x) = x2 - 4, F(x) = √x, and G(x) = 1/(x - 2). Determine the following composite functions and give their domains.g º g
More composite functions Let f (x) = x, g(x) = x2 - 4, F(x) = √x, and G(x) = 1/(x - 2). Determine the following composite functions and give their domains.F º g º g
More composite functions Let f (x) = x, g(x) = x2 - 4, F(x) = √x, and G(x) = 1/(x - 2). Determine the following composite functions and give their domains.g º g º f
More composite functions Let f (x) = x, g(x) = x2 - 4, F(x) = √x, and G(x) = 1/(x - 2). Determine the following composite functions and give their domains.f º g º G
Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals h(x) = (x³ - 5) ¹0 3 10
Composite functions and notation Let f (x) = x2 - 4, g (x) = x3, and F(x) = 1/ (x - 3). Simplify or evaluate the following expressions. f(√x + 4)
Let f (x) = x2 - 4,g (x) = x3, and F(x) = 1/ (x - 3). Simplify or evaluate the following expressions. g (F(f(x)))
Let f (x) = x2 - 4, g (x) = x3, and F(x) = 1/ ( x - 3). Simplify or evaluate the following expressions. F(F)( x))
Let f(x) = x2 - 4, g (x) = x3, and F(x) = 1/ ( x - 3) . Simplify or evaluate the following expressions. f(2+h)-f(2) h
Let ƒ(x) = x2 - 4, g(x) = x3, and F(x) = 1/ ( x - 3). Simplify or evaluate the following expressions.F(g(y)).
Graph each function with a graphing utility using the given window. Then state the domain and range of the function. g(t) 1 +72² [-7,7] × [0, 1.5] 1 1
Let ƒ(x) = x2 - 4, g(x) = x3, and F(x) = 1/ ( x - 3) . Simplify or evaluate the following expressions.F(y4)
Sketch a graph of an odd function and give the function’s defining property. y, A B X
Sketch a graph of an even function and give the function’s defining property. У 0 B A X
If ƒ(x) = 1/(x3 + 1), what is ƒ(2)? What is ƒ(y)2?
Does the independent variable of a function belong to the domain or range? Does the dependent variable belong to the domain or range?
Determine the following indefinite integrals. S₁V dx 8 xV4 x²
Determine the following indefinite integrals. ex 36-e²r a dx, x < ln 6
Evaluate the following integrals. ~3/4 sinh'x : dx 5/12 √x² + 1 V
Determine the following indefinite integrals. Livi dx xV1 + x4
Evaluate the following derivatives. f(x) = x sinh 'x − Vn +1 - X
Determine the following indefinite integrals. dx √16 + x²
Evaluate the following derivatives. f(x) = csch ¹ (2/x)
Evaluate the following derivatives. f(v) = sinh-1,2
Determine the following indefinite integrals. 15 dx x2¹x > 2√₂ 8x²
Evaluate the following derivatives. f(t) = 2 tanh~! Vì Vĩ
Evaluate the following derivatives. f(u) = sinh" (tan u)
Evaluate the following derivatives. f(x) = cosh¹ 4x
Evaluate each definite integral. . In 2 S." tanh x dx
Evaluate each definite integral. S sech² √x Vx dx
Evaluate each definite integral. In 3 In 2 csch x dx
Verify the following identities.sinh(cosh-1 x) =√x2 - 1, for x ≥1
Determine each indefinite integral. [sec sech² x tanh x dx
Determine each indefinite integral. sinh→x dx
Determine each indefinite integral. [co cosh 2x dx
Determine each indefinite integral. Itanh² tanh2x dx
Determine each indefinite integral. Ico coth²x csch² x dx
Determine each indefinite integral. S₁ sinh x 1 + cosh - dx x
Compute dy/dx for the following functions. y = Vcoth 3x
Compute dy/dx for the following functions.y = x>/csch x
Compute dy/dx for the following functions.y = x2 cosh2 3x
Compute dy/dx for the following functions.y = x tanh x
Compute dy/dx for the following functions.y = tanh2 x
Compute dy/dx for the following functions.y = -sinh3 4x
Compute dy/dx for the following functions.y = cosh2 x
Compute dy/dx for the following functions.y = sinh 4x
Express sinh-1 x in terms of logarithms.
Evaluate the following limits. Use l’Hôpital’s Rule when needed. In 10 y lim y-0¹ Vy
Determine the following indefinite integrals. Check your work by differentiation. [s sec 40 tan 40 de
Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f' (x) dx. f(x) = ln (1-x)
Evaluate the following limits. lim (1-x) tan x-17 TTX 2
Evaluate the following limits. Use l’Hôpital’s Rule when needed. 3 sin 80 lim 0-0 8 sin 30
Determine the following indefinite integrals. Check your work by differentiation. [csc csc² 6x dx
Evaluate the following limits. lim (csc 6x sin 7x) x-0
Evaluate the following limits. Use l’Hôpital’s Rule when needed. lim x→1 x4x³ r3 + 3x² + 5x - 2 rẻ – 5x + 3 x² 3
Determine the following indefinite integrals. Check your work by differentiation. 1 . dy 2y
Evaluate the following limits. lim (csc (1/x) (e¹/x - 1)) X→∞
Evaluate the following limits. Use l’Hôpital’s Rule when needed. lim X→∞ In x 100 Vx
Determine the following indefinite integrals. Check your work by differentiation. [(e²t. (e²¹ + 2√1) dt
Evaluate the following limits. TT lim XT/22 sec x
Determine the following indefinite integrals. Check your work by differentiation. TV₂ 6 25x² 2
Find the value of x that maximizes u in the figure. 3 |- -x- 4 0
Evaluate the following limits. lim (sin x), x-0+ 1- x X
Evaluate the following limits. lim (x - √x² + 1) x →∞
Evaluate the following limits. Check your results by graphing. lim (1 + x) cotx x->0+
Evaluate the following limits. lim (tan (tan 0 - sec 0) 0→π/2
Evaluate the following limits. Check your results by graphing. lim (√x + 1)¹/x
Evaluate the following limits. lim (x - √x² + 4x) X-∞
Evaluate the following limits. Check your results by graphing. lim_ (In x)* x-0+
Evaluate the following limits. Check your results by graphing. lim x/x 00
Evaluate the following limits. Check your results by graphing. lim 1- 3 X x
Show that the function s(x) = 3600(60 + x)-1 gives your average speed in mi/hr if you travel one mile in x seconds more or less than 60 mi/hr.
Determine the following indefinite integrals. Check your work by differentiation. 10t5 - 3 [101³ dt
Evaluate the following limits. Check your results by graphing. 2 lim X→∞TT 8x tan-1 X x
Evaluate the following limits. Check your results by graphing. sin #X lim (x - 1) sin x-1
For the following functions f, find the antiderivative F that satisfies the given condition. f(v) = sec v tan v; F(0) = 2
Make a complete graph of the following functions. a If an interval is not specified, graph the function on its domain .Use analytical methods and a graphing utility together in a complementary way.
Evaluate the following limits or explain why they do not exist. Check your results by graphing. lim (e5x + x) ¹/x x-0
Evaluate the following limits or explain why they do not exist. Check your results by graphing. lim (eax + x)/x, for a constant a x-0
Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use analytical methods and a graphing utility together in a complementary way.
Evaluate the following limits or explain why they do not exist. Check your results by graphing. lim (2x + x)¹/x, for a constant a X-0
Determine the following indefinite integrals. [1/2 dx X
Determine the following indefinite integrals. [2e² 2e²x dx
Find the solution of the following initial value problems. v'(x) = 4x¹/3 + 2x 1/3; v(8) = 40
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates. x20; 1.00001* .20.
Determine the following indefinite integrals. sec 2x tan 2x dx
Find the solution of the following initial value problems. y' (0) = √2 cos³ 0 + 1 2 cos²0 y 77 = 3
Determine the following indefinite integrals. [25 2 sec² x dx
Find the solution of the following initial value problems. u'(x) e2x + 4e³ ex -; u(In 2) = 2
Determine the following indefinite integrals. fa (1 + cos 30) de
Find the solution of the following initial value problems. y' (t) = 3 t +6; y(1) = 8
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates. In x20; In x

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