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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Sketch the graph of a function ƒ on [0, 4] with a discontinuity such that f has an absolute minimum but no absolute maximum.
Generalize Exercise 87: Show that if the horizontal line y = c intersects the graph of ƒ(x) = x2 + rx + s at two pointsFigure 26Data From Exercise 87Show that if the quadratic polynomial ƒ(x) = x2
Show, by considering its minimum, that ƒ(x) = x2 − 2x + 3 takes on only positive values. More generally, find the conditions on r and s under which the quadratic function ƒ(x) = x2 + rx + s
A cubic polynomial may have a local min and max, or it may have neither (Figure 27). Find conditions on the coefficients a and b ofthat ensure f has neither a local min nor a local max. 1 f(x) x) =
Show that if the quadratic polynomial ƒ(x) = x2 + rx + s takes on both positive and negative values, then its minimum value occurs at the midpoint between the two roots.
Find the min and max of ƒ(x) = xp(1 − x)q on [0, 1] where p, q > 0.
Compute dy/dx.y = sin(x + y)
Find the derivative using the appropriate rule or combination of rules. y = (9 − (5 − 2x4)7)3
Computefor the following functions:(a) ƒ(x) = x (b) ƒ(x) = x2 (c) ƒ(x) = x3Based on these examples, what do you think the limit represents? lim h→0 f(x+h) + f(x-h) - 2f(x) h²
Use the following table of values to calculate the derivative of the given function at x = 2:H(x) = ƒ(x)g(x) X f(x) g(x) 4 2 2 5 4 3 f'(x) -3 -2 g'(x) 9 3
The height at time t (in seconds) of a mass, oscillating at the end of a spring, is s(t) = 300 + 40 sin t cm. Find the velocity and acceleration at t = π/3 s.
Plot the derivative of ƒ(x) = x/(x2 + 1) over [−4, 4]. Use the graph to determine the intervals on which ƒ'(x) > 0 and f'(x) < 0. Then plot ƒ and describe how the sign of ƒ'(x) is reflected in
Use a computer algebra system to plot y2 = x3 − 4x for x and y between −4 and 4. Show that if dx/dy = 0, then y = 0. Conclude that the tangent line is vertical at the points where the curve
Use the following table of values to calculate the derivative of the given function at x = 2: x f(x) g(x) 4 2 2 5 4 3 f'(x) -3 -2 g'(x) 9 3
Find the derivative using the appropriate rule or combination of rules.y = (cos 6x + sin x2)1/2
A projectile is launched from ground level with an initial velocity ν0 at an angle θ, where 0 ≤ θ ≤ π/2. Its horizontal range is R = (2v20/g) sinθ cos θ, where g = 32 ft/s2 with ν0 in
Find the derivative using the appropriate rule or combination of rules. y = (x + 1)¹/2 x + 2
Plot ƒ(x) = x/(x2 − 1) (in a suitably bounded viewing box). Use the plot to determine whether ƒ'(x) is positive or negative on its domain {x : x ≠ ± 1}. Then compute ƒ'(x) and confirm your
Use the following table of values to calculate the derivative of the given function at x = 2:G(x) = ƒ(g(x)) x f(x) g(x) 4 2 2 5 4 3 f(x) -3 -2 g'(x) 9 3
The graph of y = sin x is shown in Figure 5, along with a tangent line at x = θ. Show that if π/2 y = sin x y -KIN tan e -X
Let P = V2R/(R + r)2 as in Example 6. Calculate dP/dr, assuming that r is variable and R is constant. EXAMPLE 6 Power Delivered by a Battery The power that a battery supplies to an apparatus such as
Show that for all points P on the graph in Figure 12, the segments O̅P̅ and P̅R̅ have equal length. y 0 Tangent line P R X
First compute y' and y" by implicit differentiation. Then solve the given equation for y, and compute y' and y" by direct differentiation. Finally, show that the results obtained by each approach are
Find the derivative using the appropriate rule or combination of rules.y = tan3 x + tan(x3)
Use the following table of values to calculate the derivative of the given function at x = 2:F(x) = ƒ(g(2x)) x f(x) g(x) 4 2 2 5 4 3 f'(x) -3 -2 g'(x) 9 3
Use the limit definition of the derivative and the addition law for the cosine function to prove that (cos x)'= − sin x.
Determine a and b such that p(x) = x2 + ax + b satisfies p(1) = 0 and p'(1) = 4.
First compute y' and y" by implicit differentiation. Then solve the given equation for y, and compute y'and y"by direct differentiation. Finally, show that the results obtained by each approach are
Use the following table of values to calculate the derivative of the given function at x = 2:K(x) = ƒ(x2) x f(x) g(x) 4 2 2 5 4 3 f'(x) -3 -2 g'(x) 9 3
Find the derivative using the appropriate rule or combination of rules.y =√4 − 3 cos x
Use the addition formula for the tangentto compute (tan x) directly as a limit of the difference quotients. You will also need to show that tan(x + h) = tan x + tan h 1 + tan x tan h
Find the derivative using the appropriate rule or combination of rules. y = z+1 z-1 N
Use implicit differentiation to calculate higher derivatives.Consider the equation y3 − 3/2 x2 = 1.(a) Show that y'= x/y2 and differentiate again to show that(b) Express y" in terms of x and y
Use the following table of values to calculate the derivative of the given function at x = 2:Find the points on the graph of ƒ(x) = x3 − 3x2 + x + 4 where the tangent line has slope 10. x f(x)
Verify the following identity and use it to give another proof of the formula (sin x)' = cos x:Use the addition formula for sine to prove that sin(a + b) − sin(a − b) = 2 cos a sin b. 2 cos (x +
Let ƒ(x) = x3 − 3x + 1. Show that ƒ'(x) ≥ −3 for all x and that, for every m > −3, there are precisely two points where ƒ'(x) = m. Indicate the position of these points and the
Find the derivative using the appropriate rule or combination of rules.y = (cos3 x + 3 cos x + 7)9
Show that the tangent lines to y = 1/3 x3 − x2 at x = a and at x = b are parallel if a = b or a + b = 2.
Use the method of the previous exercise to show that y"= −y−3 on the circle x2 + y2 = 1.
Find the points on the graph of x2/3 + y2/3 = 1 where the tangent line has slope 1.
Show that a nonzero polynomial function y = ƒ(x) cannot satisfy the equation y'= −y. Use this to prove that neither ƒ(x) = sin x nor ƒ(x) = cos x is a polynomial. Can you think of another way to
In Exercises 57–62, each limit represents a derivative ƒ'(a). Find ƒ(x) and a. -1-4 x lim 1 x-
Find the derivative using the appropriate rule or combination of rules. y = cos(1 + x) 1 + cos x
Use implicit differentiation to calculate higher derivatives.Calculate y" at the point (1, 1) on the curve xy2 + y − 2 = 0 by the following steps:(a) Find y' by implicit differentiation and
Find a such that the tangent lines to y = x3 − 2x2 + x + 1 at x = a and x = a + 1 are parallel.
Let ƒ(x) = x sin x and g(x) = x cos x.(a) Show that ƒ(x) = g(x) + sin x and g'(x) = −ƒ(x) + cos x.(b) Verify that ƒ'(x) = −ƒ(x) + 2 cos x and g'(x) = −g(x) − 2 sin x.(c) By further
Find the derivative using the appropriate rule or combination of rules.y = sec(√t2 − 9)
Use the table to compute the average rate of change of candidate A’s percentage of votes over the intervals from day 20 to day 15, day 15 to day 10, and day 10 to day 5. If this trend continues
Use implicit differentiation to calculate higher derivatives.Use the method of the previous exercise to compute y" at the point (1, 1) on the curve x3 + y3 = 3x + y − 2.Data From Exercise
Figure 6 shows the geometry behind the derivative formula (sin θ)= cos θ. Segments BA and BD are parallel to the x- and y-axes. Let Δ sin θ = sin(θ + h) − sin θ. Verify the following
Let ƒ(x) = g(x) = x. Show that (ƒ /g)' ≠ ƒ' /g'.
Explore the radius of curvature of curves. There can be many circles that are tangent to a curve at a particular point, but there is one that provides a “best fit” (Figure 13). This circle is
Find the derivative using the appropriate rule or combination of rules.y = cot7(x5)
Calculate y".y = 12x3 − 5x2 + 3x
Use the Product Rule to show that (ƒ2) = 2ƒƒ'.
Find the derivative using the appropriate rule or combination of rules. y = cos(1/x) 1 + x²
Explore the radius of curvature of curves. There can be many circles that are tangent to a curve at a particular point, but there is one that provides a “best fit” (Figure 13). This circle is
Calculate y".y = x−2/5
Show that (ƒ3)'= 3 ƒ2 ƒ'.
Find the derivative using the appropriate rule or combination of rules.y = (1 + cot5(x4 + 1))9
The brightness b of the sun (in watts per square meter) at a distance of d meters from the sun is expressed as an inverse-square law in the form b = L/4πd2, where L is the luminosity of the sun and
Calculate y".y = √2x + 3
Let ƒ, g, h be differentiable functions. Show that (ƒ gh)'(x) is equal toWrite ƒ gh as ƒ (gh). f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)
Find the derivative using the appropriate rule or combination of rules.y = √cos 2x + sin 4x
x and y are functions of a variable t. Use implicit differentiation to express dy/dt in terms of dx/dt, x, and y.x3 − 6xy2 = x
Calculate y".y = 4x/x + 1
Find the derivative using the appropriate rule or combination of rules.y = (1 − csc2(1 − x3))6
x and y are functions of a variable t. Use implicit differentiation to express dy/dt in terms of dx/dt, x, and y.y4 + 2x2 = xy
Calculate y".y = tan(x2)
Find the derivative using the appropriate rule or combination of rules.y = sin(√sin θ + 1)
The volume V and pressure P of gas in a piston (which vary in time t) satisfy PV3/2 = C, where C is a constant. Prove thatThe ratio of the derivatives is negative. Could you have predicted this from
Calculate y".y = sin2(4x + 9)
Use the limit definition of the derivative to prove the following special case of the Product Rule: d dx -(xf(x)) = f(x) + xf'(x
Find the derivative using the appropriate rule or combination of rules.y = (x + 1/x)−1/2
Show that if P lies on the intersection of the two curves x2 − y2 = c and xy = d (c, d constants), then the tangents to the curves at P are perpendicular.
Compute dy/dx.x3 − y3 = 4
The lemniscate curve (x2 + y2)2 = 4(x2 − y2) was discovered by Jacob Bernoulli in 1694, who noted that it is “shaped like a figure 8, or a knot, or the bow of a ribbon.” Find the coordinates
Find the derivative using the appropriate rule or combination of rules.y = sec(1 + (4 + x)−3/2)
Compute dy/dx.4x2 − 9y2 = 36
The vapor pressure of water at temperature T (in kelvins) is the atmospheric pressure P at which no net evaporation takes place. Use the following table to estimate the indicated derivatives using
Find the derivative using the appropriate rule or combination of rules. y = 1 + 1 + √√x
Divide the curve in Figure 15 into five branches, each of which is the graph of a function. Sketch the branches. N. 2 -2 y 2 +x 4
Compute dy/dx.y = xy2 + 2x2
Compute dy/dx.y/x = x + y
Find the derivative using the appropriate rule or combination of rules. y = √x+1+1
Exercises 72 and 73: A basic fact of algebra states that c is a root of a polynomial ƒ if and only if ƒ(x) = (x − c)g(x) for some polynomial g. We say that c is a multiple root if ƒ(x) = (x −
Exercises 72 and 73: A basic fact of algebra states that c is a root of a polynomial f if and only if ƒ(x) = (x − c)g(x) for some polynomial g. We say that c is a multiple root if ƒ(x) = (x −
Find the derivative using the appropriate rule or combination of rules.y = (kx + b)−1/3; k and b any constants
Sketch the graph of ƒ(x) = x |x|. Then show that ƒ'(0) exists.
Figure 7 is the graph of a polynomial with roots at A, B, and C. Which of these is a multiple root? Explain your reasoning using Exercise 72.Data From Exercise 72Exercises 72 and 73: A basic fact of
Compute dy/dx.tan(x + y) = xy
Find the derivative using the appropriate rule or combination of rules. y = 1 Vkt + b =;k,b constants, not both zero
Compute the higher derivative. d² dx² -sin(x²)
Compute dy/dx and d2y/dx2 .x2 − 4y2 = 8
Compute dy/dx and d2y/dx2 .6xy + y2 = 10
A light moving at 0.8 m/s approaches a man standing 4 m from a wall (Figure 10). The light is 1 m above the ground. How fast is the tip P of the man’s shadow moving when the light is 7 m from the
From a 2005 study by the Fisheries Research Services in Aberdeen, Scotland, we infer that the average length in centimeters of the species Clupea harengus (Atlantic herring) as a function of age t
The function L(t) = 12 + 3.4 sin( 2π/365 t) models the length of a day from sunrise to sunset in Sapporo, Japan, where t is the day in the year after the spring equinox on March 21. Determine L'(t),
According to a 1999 study by Starkey and Scarnecchia, the average weight (in kilograms) at age t (in years) for channel catfish in the Lower Yellowstone River (Figure 2) is approximated by the
Use the table of values to calculate the derivative of the function at the given point.g(√x), x = 16 1 4 f(x) f'(x) 5 g(x) g'(x) 545 4 6 0 6 7 4 16 3

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