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mathematics
precalculus

Questions and Answers of Precalculus

Prove the statement using the ε, δ definition of a limit. lim x' = 8
Find the limit or show that it does not exist. x* – 3x? + x lim 3 x' - x + 2
Determine the infinite limit. lim x-5+ x – 5
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.f (x) = x4
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. M(x) %3|
Find f ′(a). Зx? — 4х + 1 f(x) —
Prove the statement using the ε, δ definition of a limit. lim (x2 – 1) = 3
Find the limit or show that it does not exist. lim Vx? + 1 -2
(a) Estimate the value ofby graphing the function f (x) − (sin x)/(sin πx). State your answer correct to two decimal places.(b) Check your answer in part (a) by evaluating f (x) for values of x
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.f (x) − x3/2
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. tan x V4 - x? В(х)
(a) If G(x) − 4x2 - x3, find G′(a) and use it to find equations of the tangent lines to the curve y = 4x2 - x3 at the points (2, 8) and (3, 9).(b) Illustrate part (a) by graphing the curve and
Prove the statement using the ε, δ definition of a limit. lim (x² + 2x – 7) = 1
Find the limit or show that it does not exist. lim (Vx2 + ax Vx2 + bx
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. 1- 2t G(t) : 3 + t
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. arcsin(1 + 21) A(t) =
Prove the statement using the ε, δ definition of a limit. lim (x² 2 – 4x + 5) = 1
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. lim x? Inx
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. sin t R(t) 2 + cos Tt
Prove the statement using the ε, δ definition of a limit. lim /6 + x = 0
Find the limit or show that it does not exist. lim (/4x? х>- оо + 3х + 2х) 4x²
Find the limit or show that it does not exist. lim (/9x² + x – 3x) /9х2 + x — 3х) х>0о
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. lim x* x→0+
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Vx – 2 x' – 2 Q(x) =
Evaluate the limit, if it exists. 1 (х + h)? lim х
Find the differential of each function.(a) y = ln(sin θ)(b) y = ex/1 - ex
Find an equation of the tangent line to the curve at the given point.y = xe-x 2, (0, 0)
Find equations of the tangent lines to the curve y = x - 1/x + 1 that are parallel to the line x - 2y = 2.
A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is
Car tires need to be inflated properly because over inflation or under-inflation can cause premature tread wear. The data in the table show tire life L (in thousands of miles) for a certain type of
Differentiate.y = t3 + 3t/t2 - 4t + 3
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 g(x) = x5, then limx→2 g(x) - g(2)/x - 2
(a) Evaluate lim x→ ∞ x sin + x.(b) Evaluate lim x → 0 x sin + x.(c) Illustrate parts (a) and (b) by graphing y = x sin(1/x).
A model for the velocity of a falling object after time t is where m is the mass of the object, g = 9.8 m/s2 is the acceleration due to gravity, k is a constant, t is measured in seconds, and v
Find f(n)(x) if f (x) = 1/(2 - x).
Find an equation of the tangent line to the curve at the given point.y = sin(sin x), (π, 0)
Find the derivative of the function. Simplify where possible.F(x) = x sec-1(x3)
Find an equation of the tangent line to the curve at the given point.y = √1 + x3 , (2, 3)
If f is a differentiable function, find an expression for the derivative of each of the following functions.(a) y = x2 f (x)(b) y = f (x)/x2(c) y = x2/f (x)(d) y = 1 + xf (x)/√x
Find the derivative of the function. Simplify where possible.g(x) = arcco√x
The number of tree species S in a given area A in the Pasoh Forest Reserve in Malaysia has been modeled by the power functionS(A) = 0.882A0.842where A is measured in square meters. Find S′(100) and
Find the given derivative by finding the first few derivatives and observing the pattern that occurs.d35/dx35 (x sin x)
If g(θ) = θ sin θ, find g'' (π/6).
Find an equation of the tangent line to the curve at the given point.y = 2x, (0, 1)
Biologists have proposed a cubic polynomial to model the length L of Alaskan rock fish at age A:where L is measured in inches and A in years. Calculateand interpret your answer. 0.372A? + 3.95A +
Find y9 and y99.y = eex
Find the derivative of the function. Simplify where possible.y = tan-1(x2)
The equation of motion of a particle is s = t4 - 2t3 + t2 - t, where s is in meters and t is in seconds.(a) Find the velocity and acceleration as functions of t.(b) Find the acceleration after 1
Let P(x) − F(x)G(x) and Q(x) − F(x)yG(x), where F and G are the functions whose graphs are shown.(a) Find P'(2).(b) Find Q'(7). F1
Differentiate.f (θ) = θ cos θ sin θ
Find the limit. sin(x – 1) lim x→1 x? + x – 2
Calculate y'.y = sin2(cos√sin πx)
Calculate y'.y = cos (e√tan 3x)
Find y9 and y99.y = √1 - sec t
Find the limit. 1 - tan x lim x→T/4 sin x cos x
Calculate y'.y = x tanh -1√x
Find y9 and y99.y = 1/(1 + tan x)2
Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f′, and f′′.f (x) = ex - x3
If f (2) = 10 and f'(x) = x2 f(x) for all x, find f''(2).
The Power Rule can be proved using implicit differentiation for the case where n is a rational number, n = p/q, and y = f (x) = xn is assumed beforehand to be a differentiable function. If y = xp/q,
Find the limit. sin(x?) lim X-
The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equationy = 211.49 - 20.96 cosh 0.03291765x for the central curve of the arch, where x and y are
Differentiate the function.R(a) = (3a + 1)2
Find the derivative of the function.f (t) = eat sin bt
Calculate y'.y + x cos y = x2y
Find dy/dx by implicit differentiation.ex/y = x - y
Calculate y'.y = cosh-1(sinh x)
Find y9 and y99.y = cos (sin 3θ)
Prove the identity.sinh 2x = 2 sinh x cosh x
Find the limit. lim cos O 202
(a) Find the differential dy and (b) Evaluate dy for the given values of x and dx.y = ex/10, x = 0, dx = 0.1
Calculate y'.y = ln |x2 - 4/2x + 5|
In a murder investigation, the temperature of the corpse was 32.5°C at 1:30 pm and 30.3°C an hour later. Normal body temperature is 37.0°C and the temperature of the surroundings was 20.0°C. When
Find the derivative of the function.y = [x + (x + sin2x)3]4
Find the first and second derivatives of the function. G(r) = \F + r
A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?
At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 pm?(a) What quantities
Differentiate.y = 1/t3 + 2t2 - 1
Differentiate the function. h(t) = Vi – 4e'
If h(2) = 4 and h'(2) = -3, find (2). h(x) dx I-2
Find the derivative of the function.g(x) = ex2 - x
Show that the sum of the x- and y-intercepts of any tangent line to the curve √x + √y = √c is equal to c.
Calculate y'.y = (u - 1/u2 + u + 1)4
Find dy/dx by implicit differentiation.xy = √sx2 + y2
Find the limit.lim x→0 csc x sin(sin x)
Show that d/dx 4√1 + tanh x/1- tanh x = 1/2ex/2.
Prove the identity.cosh 2x = cosh2x + sinh2x
Calculate y'.y = ln(cosh 3x)
Find the derivative of the function.y = cos√sin (tan π x)
Find the first and second derivatives of the function.f (x) = 0.001x5 - 0.02x3
Find an equation of the tangent line to the hyperbolaat the point (x0, y0).so the four points are y? b2 .2 and y? = 75 25 16 16
(a) Find the differential dy and (b) Evaluate dy for the given values of x and dx.y = cos πx, x = 1/3 , dx = - 0.02
Find the limit. sin 0 lim 0→0 0 + tan 0
Differentiate.y = ep(p + p√p)
Find the derivative of the function.f (x) = (2x - 3)4 (x2 + x + 1)5
Find the derivative. Simplify where possible.y = coth-1 (sec x)
Calculate y'.y = sin mx/x
Find the derivative of the function.y = 234x
Suppose that f (4) = 2, g(4) = 5, f'(4) = 6, and g'(4) = - 3. Find h'(4).(a) h(x) = 3f (x) + 8g(x)(b) h(x) = f (x) g(x)(c) h(x) = f (x) g(x)(d) h(x) = g(x) f (x) + g(x)
Differentiate the function. S(p) = /p – p VP

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