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mathematics
precalculus

Questions and Answers of Precalculus

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) = 3x - 8
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 is continuous at a, then f is
Find the limit or show that it does not exist. 1) (2x? + 1)? lim (x – 1)*(x² + x)
Calculate y'.y = 10tan πθ
Prove the statement using the ε, δ definition of a limit. x2 – 2x – 8 lim
Evaluate the limit, if it exists. 4u + 1 lim и — 2 3 и>2
Explain why the function is discontinuous at the given number a. Sketch the graph of the function. 2x? — 5х — 3 х — 3 if x + 3 f(x) = a = 3 if x = 3 6.
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). (2 + h) – 32 lim h = +0.5, ±0.1, ±0.01, ±0.001, ±0.0001
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) − mx + b
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′ (r) exists, then limx → r f (x) = f
Find the derivative. Simplify where possible.G(t) = sinh (ln t)
Prove the statement using the ε, δ definition of a limit. 9 – 4x2 lim x--15 3 + 2x 9 =
Find the derivative of the function.U(y) = (y4 + 1/y2 + 1)5
Evaluate the limit, if it exists. lim t2 + t
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.g(t) = 1/√t
Find equations of the tangent line and normal line to the given curve at the specified point.y = 2x/x2 + 1, (1, 1)
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) − x2 - 2x3
Find an equation of the tangent line to the curve at the given point.y = 2ex + x, (0, 2)
Evaluate the limit, if it exists. V1 + t - V1 – t lim
Evaluate the limit, if it exists. 3 х lim х>3 х — 33
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.f (t) − 2.5t2 + 6t
According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?
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. In x – In 4 lim х — 4
Find the limit or show that it does not exist. 1 + 4x6 lim 2 – x3
For what values of x does the graph of f have a horizontal tangent?f (x) = ex cos x
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. d²y dx2 dy dx
Prove the statement using the ε, δ definition of a limit. lim x = a
Evaluate the limit, if it exists. (3 + h)¯ – 3¬' lim
Find the derivative. Simplify where possible.F(t) = ln(sinh t)
Prove the statement using the ε, δ definition of a limit. lim x? = 0
Calculate y'.y = ln |sec 5x + tan 5x|
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| is continuous at a, so is f.
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 is continuous at a, so is |f|.
Find the derivative of the function.G(x) = 4C/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) − 4 + 8x - 5x2
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. 1 +p° lim p>-1 1 + p 15
Find an equation of the tangent line to the curve at the given point.y = 2x3 - x2 + 2, (1, 3)
Find the limit or show that it does not exist. /1 + 4x6 lim 2 – x3 x -00
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.The equation x10 - 10x2 + 5 = 0 has a root in
Find the derivative. Simplify where possible.h(x) = sinh(x2)
Calculate y'.y = ecos x + cos(eX)
Find the derivative of the function.F(t) = t - √t3 + 1
Find an equation of the tangent line to the given curve at the specified point.y = 1 + x/1 + ex, (0, 1/2)
Differentiate the function.y = ex +1 + 1
Prove the statement using the ε, δ definition of a limit. lim c = c
How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?
Suppose f (π/3) = 4 and f'(π/3) = -2, and let g(x) = f(x) sin x and h(x) = (cos x)/f (x). Find (a) g'(π/3)(b) h'(π/3)
Find the derivative. Simplify where possible.g(x) = sinh2x
Find the derivative. Simplify where possible.f(x) = tanh √x
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). est – 1 t = +0.5, ±0.1, ±0.01, ±0.001, ±0.0001 lim
Calculate y'.y = x tan-1(4x)
Find the derivative of the function.F(t) = et sin 2t
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 limx → 0 f (x) = ∞ and limx → 0 g(x)
Differentiate the function. У + Be ую ,10 У
Explain, in terms of linear approximations or differentials, why the approximation is reasonable.1/9.98 ≈ 0.1002
Find the derivative. Simplify where possible.f (x) = ex cosh x
Calculate y'.y = (x2 + 1)4/(2x + 1)3 (3x - 1)5
Find the derivative of the function.J(θ) = tan2(nθ)
Explain why the function is discontinuous at the given number a. Sketch the graph of the function. x + 3 if x < -1 f(x) 2* if x > -1 a = -1
Find f'(x) and f''(x).f(x) − x/x2 - 1
Evaluate the limit, if it exists. x + 2 lim x³ + 8 x→-2
Evaluate the limit, if it exists. (2 + h)³ – 8 lim
Find the limit. lim e*
Differentiate the function. .2 1 + 16t2 D(t) (41)
Find the limit or show that it does not exist. 4x3 + 6х? — 2 lim х> — оо 2x3 — 4х + 5
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 is continuous on [21, 1] and f (-1) = 4
Explain why the function is discontinuous at the given number a. Sketch the graph of the function. if x + -2 x + 2 f(x) = a = -2 if x = -2
Evaluate the limit, if it exists. (-5 + h)? – 25 25 lim
If f(t) = sec t, find f''(π/4).
Find the limit or show that it does not exist. lim x→-0 x +1
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 is continuous at 5 and f (5) = 2 and f
Evaluate the limit, if it exists. 2x? + 3x + 1 lim x² – 2x – 3 2.x
The graph of a function f is shown.(a) Find the average rate of change of f on the interval [20, 60].(b) Identify an interval on which the average rate of change of f is 0.(c) Which interval gives a
Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9. y. у%3D 4х — 5 7+e 7- 8 х 3 - 8 3+8 lim (3x + 5) = -1
Explain, in terms of linear approximations or differentials, why the approximation is reasonable.√4.02 ≈ 2.005
Calculate y'.y = ln sin x - 1/2 sin2x
Find the derivative of the function.H(r) = (r2 - 1)3 (2r + 1)5
Differentiate the function. νυ-2ve' 2υε f(v) υ
Calculate y'.y = (cos x)x
Find the derivative of the function.f (z) = ez/(z-1)
Find f'(x) and f''(x).f (x) = √x ex
Differentiate the function. A + Bz + Cz² F(2) : z2
(a) If f (x) − ex cos x, find f'(x) and f''(x).(b) Check to see that your answers to part (a) are reasonable by graphing f , f', and f''.
Find the limit. lim (Vx? + 4x + 1– x)
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.x2 1 2xy 1 4y2 = 12, s2, 1d (ellipse)
Use a linear approximation (or differentials) to estimate the given number.cos 29°
Find the limit or show that it does not exist. 1 – x? lim r0 x - x + 1 3
Find the limit. 1- 2x? – x* 2x? — lim 5 + x – 3x* x -00
Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9. y. у%3D 4х — 5 7+e 7- 8 х 3 - 8 3+8 lim (2x – 5) = 3 x→4
The displacement (in feet) of a particle moving in a straight line is given by s − 1/2 t2 - 6t + 23, where t is measured in seconds.(a) Find the average velocity over each time interval:(i) [4,
Calculate y'.y = log5(1 + 2x)
Find the derivative of the function.r(t) = 102√t
Differentiate the function.G(q) = (1 + q-1)2
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 (1) > 0 and f (3) < 0, then there
Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. g(x) 3x + 6' (-0, -2)
Find the limit or show that it does not exist. Зх — 2 lim x0 2x + 1
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.x2 - xy - y2 = 1, (2, 1) (hyperbola)
Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. |f(x) = x + Vx – 4, [4, 0) /x – 4
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.A function can have two different horizontal
Use a linear approximation (or differentials) to estimate the given number.e0.1

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