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mathematics
calculus 4th

Questions and Answers of Calculus 4th

In Exercises 21–32, calculate the derivative. h(t) = 6 √t + 1 √t
In Exercises 21–32, calculate the derivative.ƒ(s) = 4√s + 3√s
In Exercises 21–32, calculate the derivative.W(y) = 6y4 + 7y2/3
In Exercises 21–32, calculate the derivative.g(x) = π2
In Exercises 21–32, calculate the derivative.ƒ(x) = xπ
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
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
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
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 +
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 +
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
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
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 =
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
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
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
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
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
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) +
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
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
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
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
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
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
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
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
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
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
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,
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.
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

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