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study help
mathematics
precalculus

Questions and Answers of Precalculus

Find dy/dx by implicit differentiation.tan-1(x2y) = x + xy2
Prove the identity.tanh(ln x) = x2 - 1/x2 + 1
Find the limit. sin 3x sin 5x lim .2
Find the derivative. Simplify where possible.y = sech-1 (e-x)
Calculate y'.y = x sinh(x2)
Find the derivative of the function.g(x) = (2rarx + n)p
Find the limit.lim x → 0 sin 3x/5x3 - 4x
Find the derivative. Simplify where possible.y = x sinh-1 (x/3) - √9 + x2
Calculate y'.y = (x + λ)4/x4 + λ4
(a) Find the differential dy and(b) Evaluate dy for the given values of x and dx.y = √3 + x2 , x = 1, dx = -0.1
Differentiate.h(r) = aer b + er
Evaluate - 1 sin x lim х т
(a) If f (x) = sec x - x, find f'(x).(b) Check to see that your answer to part (a) is reasonable by graphing both f and f' for |x | < π/2.
Find the derivative of the function. y = Vx + Vx + I
Prove that d/dx (sec x) = sec x tan x.
Find the derivative of the function.g(x) = (x2 + 1)3 (x2 + 2)6
Calculate y'.y = cot(csc x)
Find dy/dx by implicit differentiation.x sin y + y sin x = 1
Find f′ (x). Compare the graphs of f and f′ and use them to explain why your answer is reasonable.f (x) = x5 - 2x3 + x - 1
If t(x) − x/ex, find g(n)(x).
Find the limit.lim → 0 cos θ - 1/sin θ
Find the derivative. Simplify where possible.y = x tanh-1x + ln √1 - x2
Prove the identity.1 + tanh x / 1 - tanh x = e2x
Calculate y'.y = √x + 1 (2 - x)5/(x + 3)7
Find the derivative of the function..f (t) = sin2(esin2 t )
Find the limit.lim t→0 tan 6t/sin 2t
(a) Find the differential dy and(b) Evaluate dy for the given values of x and dx.y = x + 1/x - 1, x = 2, dx = 0.05
Find the derivative. Simplify where possible.y = cosh-1√x
Calculate y'.xey = y - 1
Find the derivative of the function.y = esin 2x + sin(e2x)
Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.y = x - √x , (1, 0)
(a) If f (x) = (x2 - 1)ex, find f' (x) and f''(x).(b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f', and f''.
The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 15°C the pressure is 101.3 kPa at sea level and 87.14 kPa
If x2 + xy + y3 = 1, find the value of y''' at the point where x = 1.
Find the limit.lim x→0 sin x/sin πx
Find the derivative. Simplify where possible.y = sinh-1(tan x)
Calculate y'.y = tan2 (sin θ)
Differentiate.y = s - √s/s2
Find the derivative of the function.f (t) = tan(sec(cos t))
Find the derivative of the function.h(t) = (t + 1)2/3 (2t2 - 1)3
Find the limit.lim x→0 sin 5x/3x
Find the derivative. Simplify where possible.g(t) = t coth √t2 + 1
Calculate y'.y = arctan (arcin √x)
Calculate y'.y = tan(t/1 + t2)
Differentiate the function. Зе у3 * + Vх 3
Find dy/dx by implicit differentiation.sin(xy) = cos (x + y)
Prove the identity.(cosh x + sinh x)n = cosh nx + sinh nx (n any real number)
Find the numerical value of each expression.(a) sinh 0(b) cosh 0
Find y'' by implicit differentiation.x2 + 4y2 = 4
Find the derivative of the function.y = √1 + xe-2x
Find y'' by implicit differentiation.x3 - y3 = 7
Find equations of the tangent line and normal line to the curve at the given point.y2 = x3, (1, 1)
Compute Δy and dy for the given values of x and dx = Δx. Then sketch a diagram like Figure 5 showing the linesegments with lengths dx, dy, and Δy.y = x2 - 4x, x = 3, Δx = 0.5
(a) If f (x) = ex/(2x2 + x + 1), find f'(x).(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30°, with a possible error of ±1°.(a) Use differentials to estimate the error in computing the length
Find the derivative. Simplify where possible.f(t) = 1 + sinh t/1 - sinh t
Calculate y'.y = sin(tan √1 + x3)
Find the derivative of the function.y = cot2(sin θ)
Find y'' by implicit differentiation.sin y + cos x = 1
Calculate y'.y = √t ln(t4)
Find the derivative of the function.y = x2e-1/x
Find y'' by implicit differentiation.x2 + xy + y2 = 3
Find an equation of the tangent line to the curve at the given point. y = Vx - x, (1, 0)
(a) The curve y = x/(1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3, 0.3).(b) Illustrate part (a) by graphing the curve and the tangent line on the
A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 L/min. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1
Find the derivative. Simplify where possible.y = sech x (1 + ln sech x)
Find dy/dx by implicit differentiation.tan (x - y) = y/1 + x2
Evaluate the limit, if it exists. (x + h)³ – x³ lim
Evaluate the limit, if it exists. Vx? + 9 – 5 x + 4 lim
Prove the statement using the precise definition of a limit. lim (14 – 5x) = 4
Find the limit or show that it does not exist. х+ 3x2 lim 4х — 1 х> 00
Prove the statement using the ε, δ definition of a limit. lim |x| =0
If g(x) = x4 - 2, find g′(1) and use it to find an equation of the tangent line to the curve y = x4 - 2 at the point (1, -1).
Evaluate the limit, if it exists. x? — 4х + 4 lim .2 Зx? — 4 x>2 х4
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. х? — 1 f(x) : - 1 2х — 3
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 5' – 1 lim
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.g(x) − √9 - x
Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. sin 30 lim 0→0 tan 20
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. x? + 1 G(x) 2x2 — х — 1
Evaluate the limit, if it exists. 4 - Vx lim x→16 16x x?
Sketch the graph of a function f where the domain is (-2, 2),f is continuous at all numbers in its domain except ±1, and f is odd. F'(0) = -2, lim,-2-f(x) = ∞, f
Prove the statement using the ε, δ definition of a limit. lim x = 0
Find the limit or show that it does not exist. ух + 3x2 lim 4х — 1 х— оо
Find the limit or show that it does not exist. .2 lim Vx4 + 1
Find the limit. lim tan (1/x)
Find the limit or show that it does not exist. Vt + lim 2t – t? .2 t→0
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.Let f be a function such that limx→0 f (x)
Prove the statement using the ε, δ definition of a limit. 2 + 4x lim 3
For the function f graphed in Exercise 18:(a) Estimate the value of f′(50).(b) Is f′(10). f′(30)?(c) Is f′(60) > f(80) - f(40)/80 - 40? Explain.Exercise 18:The graph of a function f is
Evaluate the limit, if it exists. 14 – 1 lim →1 t³ – 1
Explain why the function is discontinuous at the given number a. Sketch the graph of the function. x? - if x + 1 x? - 1 f(x) = if x = 1
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). x? – 3x lim 3 x? - 9' x = 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95,
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f (x) > 1 for all x and limx→0 f (x)
Find the limit or show that it does not exist. t – tvE 2t3/2 + 3t – 5 lim H∞ 2t3/2
Find the limit. lim х? — Зх + 2, x-1 x - 1 х —
Prove the statement using the ε, δ definition of a limit. lim (3 – x) = -5 x- 10
Sketch the graph of a function t that is continuous on its domain (-5, 5) and where g(0) − 1, g′(0) − 1, g′(-2) − 0, limx→-5+ g(x) = ∞, and limx→-5- g(x) = 3.
Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = -3 and g′(5) = 4.
Evaluate the limit, if it exists. V9 + h – 3 lim
Explain why the function is discontinuous at the given number a. Sketch the graph of the function. if x 0 cos x f(x) = {0 a= 0

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