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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

In Exercises 21–32, calculate the derivative.h(t) =√2t √2
In Exercises 21–32, calculate the derivative. R(z) = 25/3 - 423/2 N
In Exercises 33–38, calculate the derivative by expanding or simplifying the function.P(s) = (4s − 3)2
In Exercises 33–38, calculate the derivative by expanding or simplifying the function.Q(r) = (1 − 2r)(3r + 5)
In Exercises 33–38, calculate the derivative by expanding or simplifying the function.ƒ(x) = (2 − x)(2 + x)
In Exercises 33–38, calculate the derivative by expanding or simplifying the function. g(x) = x² + 4x¹/2 +रे
In Exercises 33–38, calculate the derivative by expanding or simplifying the function.g(w) = (1 + 2w)3
In Exercises 33–38, calculate the derivative by expanding or simplifying the function. s(t) = 1 - 2t 1¹/2
In Exercises 39–44, calculate the derivative indicated. dT dC lc-8 T = 3C2/3
In Exercises 39–44, calculate the derivative indicated. dP dv\v=-2² P = 7 V $dP dv\v=-2 P = 7 V$
In Exercises 39–44, calculate the derivative indicated. ds dz z=2 Iz=2' s = 4z - 16z²
In Exercises 39–44, calculate the derivative indicated. dR dW lw=1 R = W"
In Exercises 39–44, calculate the derivative indicated. dr dt 11=4 r = 22² +1 11/2
In Exercises 39–44, calculate the derivative indicated. dp| dh \h=32 p = 16h0.2 +8h-0.8 $dp| dh \h=32 p = 16h0.2 +8h-0.8$
Match the functions in graphs (A)–(D) with their derivatives (I)–(III) in Figure 10. Note that two of the functions have the same derivative. Explain why. (A) (B) (I) X pl (C) EVN (II) (D) (III)
Of the two functions ƒ and g in Figure 11, which is the derivative of the other? Justify your answer. y 2f -f(x) -g(x) X
According to the Peak Oil Theory, first proposed in 1956 by geophysicist M. Hubbert, the total amount of crude oil Q(t) produced worldwide up to time t has a graph like that in Figure 13.(a) Sketch the derivative Q(t) for 1900 ≤ t ≤ 2150. What does Q(t) represent?(b) In which year
Prove each of the following using the definition of the derivative.(a) The First-Power Rule: d/dx x = 1(b) The Constant Rule: d/x c = 0
Compute the derivatives, where c is a constant. (a) ct3 dt d (c) (9c²y³ - 24c) dy (b) d dz (5z + -4cz²)
Find the points on the graph of ƒ(x) = 12x − x3 where the tangent line is horizontal.
Find the points on the graph of y = x2 + 3x − 7 at which the slope of the tangent line is equal to 4.
Find the values of x where y = x3 and y = x2 + 5x have parallel tangent lines.
Find all values of x such that the tangent line to y = 4x2 + 11x + 2 is steeper than the tangent line to y = x3.
In Exercises 1–6, use the Product Rule to calculate the derivative.ƒ(x) = (3x − 5)(2x2 − 3)
In Exercises 1–6, use the Product Rule to calculate the derivative.ƒ(x) = √x (1 − x3)
In Exercises 1–6, use the Product Rule to calculate the derivative.ƒ(x) = (3x4 + 2x6)(x − 2)
In Exercises 1–6, use the Product Rule to calculate the derivative.dh/ds Ι s = h(s) = (s−1/2 + 2s)(7 − s−1)
In Exercises 7–12, use the Quotient Rule to calculate the derivative. f(x) = x x-2
In Exercises 1–6, use the Product Rule to calculate the derivative.y = (t − 8t−1)(t + t2)
In Exercises 7–12, use the Quotient Rule to calculate the derivative. dg| dt 1=-2 g(t) = = 1²2 +1 1²-1
In Exercises 7–12, use the Quotient Rule to calculate the derivative. f(x) = x + 4 x² + x + 1
In Exercises 7–12, use the Quotient Rule to calculate the derivative. dw| dz |z=9 W = z² √z+z
In Exercises 7–12, use the Quotient Rule to calculate the derivative. g(x)= 1 1 + x3/2
In Exercises 7–12, use the Quotient Rule to calculate the derivative. h(s) = $3/2 s² + 1
In Exercises 13–18, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and directly calculate the derivative. h(x) = -
In Exercises 13–18, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and directly calculate the derivative.ƒ(x) = x3x−3
In Exercises 13–18, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and directly calculate the derivative.ƒ(t) = (2t + 1)(t2 − 2)
In Exercises 13–18, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and directly calculate the derivative. h(t) = 2²-1 t-1
In Exercises 13–18, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and directly calculate the derivative.ƒ(x) = x2(3 + x−1)
In Exercises 13–18, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and directly calculate the derivative. g(x) = x³ + 2x² + 3x¹ X
In Exercises 19–40, calculate the derivative.ƒ(x) = (x3 + 5)(x3 + x + 1)
In Exercises 19–40, calculate the derivative. dz dxx-- =-2 Z= X 3x² + 1
In Exercises 19–40, calculate the derivative. dy dx \x=3 y = 1 x + 10 $dy dx \x=3 y = 1 x + 10$
In Exercises 19–40, calculate the derivative.ƒ(x) = (1/x− x2) (x3 + 1)
In Exercises 19–40, calculate the derivative. f(x) = 9x5/2-2
In Exercises 19–40, calculate the derivative.ƒ(x) = (√x + 1)(√x − 1)
In Exercises 19–40, calculate the derivative. dy dx x=2 y = -4 x² - 5
In Exercises 19–40, calculate the derivative. dz dx\x=1 Z= 1 ³+1 $dz dx\x=1 Z= 1 +1$
In Exercises 19–40, calculate the derivative. f(x) = x² + x²] x + 1
In Exercises 19–40, calculate the derivative. f(x) = 3x³x²+2 √x
In Exercises 19–40, calculate the derivative. h(t)= = t (t + 1)(t² + 1)
In Exercises 19–40, calculate the derivative.ƒ(x) = x3/22x4 − 3x + x−1/2)
In Exercises 19–40, calculate the derivative.ƒ(x) = x2/3(x2 − 1)
In Exercises 19–40, calculate the derivative.h(x) = π2(x − 1)
In Exercises 19–40, calculate the derivative. g(z) = (z-2)(z² + 1) Z
In Exercises 19–40, calculate the derivative.ƒ(x) = (x + 3)(x − 1)(x − 5)
In Exercises 19–40, calculate the derivative. f(x) = 13/2(x²+1) x + 1
In Exercises 19–40, calculate the derivative.h(s) = s(s + 4)(s2 + 1)
In Exercises 19–40, calculate the derivative.Simplify first. g(z) = (2²-4) (z²-1 z-1/z+2
In Exercises 19–40, calculate the derivative. d dt xt - 4 1² - x (x constant)
In Exercises 19–40, calculate the derivative. d -((ax + b)(abx² + 1)) (a, b constants) dx
In Exercises 19–40, calculate the derivative. d (ax + b dx cx+d) (a, b, c, d constants)
In Exercises 45–48, calculate the derivative using the values: H'(4), where H(x) = X g(x)f(x)
Find all values of a such that the tangent line topasses through the origin (Figure 5). f(x) = x-1 x + 8 at x = a
Current I (amperes), voltage V (volts), and resistance R (ohms) in a circuit are related by Ohm’s Law, I = V/R. (a) Calculate (b) Calculate dI dR\R=6 dV dR R=6 if V is constant with value V = 24. if I is constant with value I = 4. $(a) Calculate (b) Calculate dI dR\R=6 dV dR R=6 if V is constant with value V = 24. if I is constant with$
The revenue per month earned by the Couture clothing chain at time t is R(t) = N(t)S (t), where N(t) is the number of stores and S (t) is average revenue per store per month. Couture embarks on a two-part campaign: (A) to build new stores at a rate of five stores per month, and (B) to use
The curve y = 1/(x2 + 1) is called the witch of Agnesi (Figure 6) after the Italian mathematician Maria Agnesi (1718– 1799). This strange name is the result of a mistranslation of the Italian word la versiera, meaning “that which turns.” Find equations of the tangent lines at x = ±1.
The tip speed ratio of a turbine is the ratio R = T/W, where T is the speed of the tip of a blade and W is the speed of the wind. (Engineers have found empirically that a turbine with n blades extracts maximum power from the wind when R = 2π/n.) Calculate dR/dt (t in minutes) if W = 35 km/h and W
Use the limit definition to proveShow that the difference quotient for 1/ ƒ(x) is equal to d 1 dx f(x), = || f'(x) f² (x)
Prove the Quotient Rule using the limit definition of the derivative.
Prove the Quotient Rule using Eq. (1) and the Product Rule. -g(x) + lim f(x + h) 8(x + h) − g(x) h→0 h We show that this equals f(x)g'(x). (fg)'(x) = lim f(x+h)-f(x) h h→0 We show that this equals f'(x)g(x).
Use the limit definition of the derivative to prove the following special case of the Quotient Rule: d (f(x) dx X = xf'(x) = f(x) zx
If you are familiar with proof by induction, use induction to prove the Power Rule for all whole numbers n. Show that the Power Rule holds for n = 1; then write xn as x · xn−1 and use the Product Rule.
Compute the derivative of ƒ(x) = x3/2 using the limit definition. Show that f(x+h)— f(x) _ (x+h)3 – x3 ( h h 1 (x+h) + (x3
Compute the derivative of ƒ(x) = x1/3 using the limit definition. Multiply the numerator and denominator in the difference quotient ƒ(x + h) − ƒ(x) h by ε/2*² + ε/1¹* € / 1 (y + x) + €/z(y + x)
The average speed (in meters per second) of a gas molecule iswhere T is the temperature (in kelvins), M is the molar mass (in kilograms per mole), and R = 8.31. Calculate dvavg/dT at T = 300 K for oxygen, which has a molar mass of 0.032 kg/mol. Vavg || 8RT ẨM
Show, using the limit definition of the derivative, that ƒ(x) = |x2 − 4| is not differentiable at x = 2.
The Clausius–Clapeyron Law relates the vapor pressure of water P (in atmospheres) to the temperature T (in kelvins):Do your estimates seem to confirm the Clausius–Clapeyron Law? What is the approximate value of k? T (K) P (atm) dP dT dP dT ≈ = k where k is a constant. Estimate dP/dT for T =
A power law model relating the kidney mass K in mammals (in kilograms) to the body mass m (in kilograms) is given by K = 0.007m0.85. Calculate dK/dm at m = 68. Then calculate the derivative with respect to m of the relative kidney-to-mass ratio K/m at m = 68.
Let L be the tangent line to the hyperbola xy = 1 at x = a, where a > 0. Show that the area of the triangle bounded by L and the coordinate axes does not depend on a.
Match functions (A)–(C) with their derivatives (I)–(III) in Figure 15. (A) A (B) S.. (C) X th (I) (II) X (III) X A.
In the setting of Exercise 68, show that the point of tangency is the midpoint of the segment of L lying in the first quadrant.Data From Exercise 68Let L be the tangent line to the hyperbola xy = 1 at x = a, where a > 0. Show that the area of the triangle bounded by L and the coordinate axes does
Make a rough sketch of the graph of the derivative of the function in Figure 16(A). 1 + 2 (A) + 3 4 -X
Graph the derivative of the function in Figure 16(B), omitting points where the derivative is not defined. -1 3 4+ 2+ 0 y + 1 2 (B) 3 4
Determine the values of x at which the function in Figure 17 is: (a) Discontinuous and (b) Nondifferentiable. + 2 3 4 X
In Exercises 75–80, zoom in on a plot of ƒ at the point (a, ƒ (a)) and state whether or not ƒ appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist.ƒ(x) = (x − 1)|x|, a = 0
In Exercises 75–80, zoom in on a plot of ƒ at the point (a, ƒ (a)) and state whether or not ƒ appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist.ƒ(x) = (x − 3)5/3, a = 3
In Exercises 75–80, zoom in on a plot of ƒ at the point (a, ƒ (a)) and state whether or not ƒ appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist.ƒ(x) = (x − 3)1/3, a = 3
In Exercises 75–80, zoom in on a plot of ƒ at the point (a, ƒ (a)) and state whether or not ƒ appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist.ƒ(x) = sin(x1/3), a = 0
In Exercises 75–80, zoom in on a plot of ƒ at the point (a, ƒ (a)) and state whether or not ƒ appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist.ƒ(x) = |sin x|, a = 0
Find the coordinates of the point P in Figure 18 at which the tangent line passes through (5, 0). f(x)=9-x² -3 9 FIGURE 18 4 + 5 Xx
In Exercises 75–80, zoom in on a plot of ƒ at the point (a, ƒ (a)) and state whether or not ƒ appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or does not exist.ƒ(x) = |x − sin x|, a = 0
Exercises 83–86 refer to Figure 19. Length QR is called the subtangent at P, and length RT is called the subnormal.Calculate the subtangent ofƒ(x) = x2 + 3x at x = 2 y Q P=(x, f(x)) R y=f(x) Tangent line T -X
Exercises 83–86 refer to Figure 19. Length QR is called the subtangent at P, and length RT is called the subnormal.Show that for n ≠ 0, the subtangent of ƒ(x) = xn at x = c is equal to c. y Q P=(x, f(x)) R y=f(x) Tangent line T -X
Prove the following theorem of Apollonius of Perga (the Greek mathematician born in 262 BCE who gave the parabola, ellipse, and hyperbola their names): The subtangent of the parabola y = x2 at x = a is equal to a/2.
Show that the subtangent to y = x3 at x = a is equal to 1/3a.
Formulate and prove a generalization of Exercise 88 for y = xn.Data From Exercise 88Show that the subtangent to y = x3 at x = a is equal to 1/3a.
Two small arches have the shape of parabolas. The first is given by ƒ x) = 1 − x2 for −1 ≤ x ≤ 1 and the second by g(x) = 4 − (x − 4)2 for 2 ≤ x ≤ 6. A board is placed on top of these arches so it rests on both (Figure 20). What is the slope of the board? Find the tangent line to
A vase is formed by rotating y = x2 around the y-axis. If we drop in a marble, it will either touch the bottom point of the vase or be suspended above the bottom by touching the sides (Figure 21). How small must the marble be to touch the bottom?
Let n be a whole number. Calculate the derivative of ƒ(x) = x−n by showing that f(x+h)-f(x) h -1 x(x + h)" (x + h)" - xn h

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