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mathematics
calculus 4th

Questions and Answers of Calculus 4th

A 900-kg rocket is released from a space station. As it burns fuel, the rocket’s mass decreases and its velocity increases. Let v(m) be the velocity (in meters per second) as a function of mass m.
Use the Change of Variables Formula to evaluate the definite integral. S 4x + 12 (x² + 6x + 1)² dx
On a typical day, a city consumes water at the rate of r(t) = 100 + 72t − 3t2 (in thousands of gallons per hour), where t is the number of hours past midnight. What is the daily water consumption?
Prove that 0.277 = VI π/4 JA/8 cos xdx ≤ 0.363.
Use R5 and L5 to show that the area A under y = x−2 over [10, 13] satisfies 0.0218 ≤ A ≤ 0.0244.
As water flows through a tube of radius R = 10 cm, the velocity v of an individual water particle depends only on its distance r from the center of the tube. The particles at the walls of the tube
Use the Change of Variables Formula to evaluate the definite integral. S'e (x + 1)(x² + 2x)5³ dx
The learning curve in a certain bicycle factory is L(x) = 12x−1/5 (in hours per bicycle), which means that it takes a bike mechanic L(n) hours to assemble the nth bicycle. If a mechanic has
Prove that 0 VI 5/2 sin x Jx/4 dx < √2 2
Use R4 and L4 to show that the area A under the graph of y = sin x over [0, π/2] satisfies 0.79 ≤ A ≤ 1.19.
Verify the linearity properties of the indefinite integral stated in Theorem 4. THEOREM 4 Additivity for Adjacent Intervals Let a ≤ b ≤ c, and assume that f is integrable. Then [ f(x) dx = = -
Cost engineers at NASA have the task of projecting the cost P of major space projects. It has been found that the cost C of developing a projection increases with P at the rate dC/dP ≈ 21P−0.65,
Use the Change of Variables Formula to evaluate the definite integral. 17 10 (x-9)-2/3 dx
Find constants c1 and c2 such that F(x) = c1 sin 3x + c2x cos 3x is an antiderivative of ƒ(x) = 2x sin 3x.
Show that the area A under ƒ(x) = x−1 over [1, 8] satisfies + 15 16 + -100 113 + } = A = 1 + } + } + −4 116
An astronomer estimates that in a certain constellation, the number of stars per magnitude m, per degree-squared of sky, is equal to A(m) = 2.4 × 10−6m7.4 (fainter stars have higher magnitudes).
Use the Change of Variables Formula to evaluate the definite integral. π/2 L12 cos xdx -1/2 √sinx+1
Find constants c1 and c2 such that F(x) = c1x cos x + c2 sin x is an antiderivative of ƒ(x) = x sin x.
Suppose that ƒ(x) ≤ g(x) on [a, b]. By the Comparison Theorem, Is it also true that ƒ(x) ≤ g (x) for x ∈ [a, b]? If not, give a counterexample. Ja ≤ ²80 Ja f(x) dx ≤ g(x) dx.
Show that the area A under y = x1/4 over [0, 1] satisfies LN ≤ A ≤ RN for all N. Use a computer algebra system to calculate LN and RN for N = 100 and 200, and determine A to two decimal places.
Evaluate ∫8−8 x15 dx / 3 + cos2 x, using the properties of odd functions.
Use the Change of Variables Formula to evaluate the definite integral. π/6 Jo sec² (2x - 7) dx
Suppose that F (x) = ƒ(x) and G(x) = g(x). Is it true that y = F(x)G(x) is an antiderivative of y = ƒ(x)g(x)? Confirm or provide a counterexample.
State whether the following statement is true or false. If false, sketch the graph of a counterexample. (a) If f(x) > 0, then (b) If ہیں So f(x) dx > 0. f(x) dx > 0, then f(x) > 0.
Show that the area A under y = 4/(x2 + 1) over [0, 1] satisfies RN ≤ A ≤ LN for all N. Determine A to at least three decimal places using a computer algebra system. Can you guess the exact value
Evaluate ∫10 ƒ(x) dx, assuming that ƒ is an even continuous function such that Si se f(x) dx = 5, [ f(x f(x) dx = 8
Use the Change of Variables Formula to evaluate the definite integral. T/2 5*¹² cos3 x sin xdx 10
Suppose that F(x) = ƒ(x).(a) Show that y = 1/2F(2x) is an antiderivative of y = ƒ(2x).(b) Find the general antiderivative of y = ƒ(kx) for k ≠ 0.
Explain graphically: If ƒ is an odd function, then S fo -a f(x) dx = 0.
Compute R100 from Example 5, approximating the area under the graph of ƒ(x) = sin x between π/4 and 3π/4. EXAMPLE 5 Let A be the area under the graph of f(x) = sinx on the interval [4,3] (Figure
Plot the graph of ƒ(x) = sin mx sin nx on [0, π] for the pairs (m, n) = (2, 4), (3, 5) and in each case guess the value of I = ∫π0 ƒ(x) dx. Experiment with a few more values (including two
Find an antiderivative for ƒ(x) = |x|.
Use the Change of Variables Formula to evaluate the definite integral. T/2 Jπ/3 cot X 2 x csc²=dx 2 z dx
Compute (sin x)(sin2 x + 1) dx. 1 -1
Compute R100 from Exercise 67, approximating the area under the graph of ƒ(x) = sin x between 0 and π. Can you guess the exact value of the area?Data From Exercise 67Express the area under the
Show thatwhere F'(x) = ƒ(x) and G'(x) = F(x). Use this to evaluate [x x f(x) dx = xF(x) - G(x)
Using Theorem 1, prove that if F'(x) = ƒ(x), where ƒ is a polynomial of degree n − 1, then F is a polynomial of degree n. Then prove that if g is any function such that g(n)(x) = 0, then g is a
Evaluate ∫20 r√5 − √4 − r2 dr.
Let k and b be positive. Show, by comparing the right-endpoint approximations, that S S x dx = fk+1 xk dx
The Power Rule for antiderivatives does not apply to ƒ(x) = x−1. Which of the graphs in Figure 4 could plausibly represent an antiderivative of ƒ(x) = x−1? 3 (A) +x 5 3 (B) + 5 E 3 (C) 5 x
Compute R100, approximating the area under the graph of ƒ(x) = 1/x between 1 and 2.
Suppose that ƒ and g are continuous functions such that, for all a,Give an intuitive argument showing that ƒ(0) = g(0). Explain your idea with a graph. S f(x) dx = -a -a g(x) dx
Separately for n = 2, 3, 4, and 9, compute R100, approximating the area under the graph of ƒ(x) = xn between 0 and 1. Can you guess what the area is in general, expressed in terms of n?
Plot the graph of ƒ(x) = x−2 sin x, and show that 0.2 ≤ ∫21 ƒ(x) dx ≤ 0.9.
Theorem 4 remains true without the assumption a ≤ b ≤ c. Verify this for the cases b THEOREM 4 Additivity for Adjacent Intervals Let a ≤ b ≤ c, and assume that fis integrable. Then = [²
Evaluate ∫π/20 sinn x cos x dx for n ≥ 0.
Find upper and lower bounds for ∫10 ƒ(x) dx, for y = ƒ(x) in Figure 7. 2+ 1 y=x² + 1 1 y = f(x) y=x¹/2 + 1
Although the accuracy of RN generally improves as N increases, this need not be true for small values of N. Draw the graph of a positive continuous function ƒ on an interval such that R1 is closer
Draw the graph of a positive continuous function on an interval such that R2 and L2 are both smaller than the exact area under the graph. Can such a function be monotonic?
Use substitution to evaluate the integral in terms of ƒ(x). S₁ f(x)³ f'(x) dx
Use substitution to evaluate the integral in terms of ƒ(x). - f'(x) f(x)² dx
Explain graphically: The endpoint approximations are less accurate when ƒ(x) is large.
Prove that for any function ƒ on [a, b], RN - LN = b-a N -(f(b)-f(a))
Use substitution to evaluate the integral in terms of ƒ(x). f'(x) f(x) S dx
In this exercise, we prove that exist and are equal if ƒ is increasing (the case of ƒ decreasing is similar). We use the concept of a least upper bound discussed in Appendix B. lim Ry and lim
Use substitution to evaluate the integral in terms of ƒ(x). Ss f'(-x + 7) dx
Use Eq. (1) to show that if f is positive and monotonic, then the area A under its graph over [a, b] satisfiesEquation (1) \RN - A| ≤ b = a N |f(b)-f(a)|
Use Eq. (2) to find a value of N such that |RN − A| −4 for the given function and interval.Equation 2 f(x)=√x, [1,4]
Show that ∫π/60 ƒ(sin θ) dθ = ∫1/20 ƒ(u) 1/√1 − u2 du.
Use the substitution u = 1 + x1 to show thatEvaluate for n = 2, 3. S Su √1 + x¹/n dx = n u¹/² (u - 1)"-¹ du
Explain with a graph: If ƒ is increasing and concave up on [a, b], then LN is more accurate than RN. Which is more accurate if f is increasing and concave down?
Use Eq. (2) to find a value of N such that |RN − A| −4 for the given function and interval.Equation 2 f(x) = √√9-x², [0,3]
Explain with a graph: If ƒ is linear on [a, b], then the * f(x) dx = (RN + LN) for all N. =
Prove that if ƒ is positive and monotonic, then MN lies between RN and LN and is closer to the actual area under the graph than both RN and LN. In the case that ƒ is increasing, Figure 18 shows
Use substitution to prove that ∫a−a ƒ(x) dx = 0 if ƒ is an odd function.
LetProve that F(x) and y = cosh−1 x differ by a constant by showing that they have the same derivative. Then prove they are equal by evaluating both at x = 1. F(x)=x√x²-1-2 -²5₁
Prove that Then show that the regions under the hyperbola over the intervals [1, 2], [2, 4], [4, 8], . . . all have the same area (Figure 3). bla 1 1 L² = = dx = fi = = dx te - Ja x X dx for a, b 0.
Let f be a positive increasing continuous function on [a, b], where 0 ≤ a 1 = bf(b) - af (a) - * f(x) dx f(b) f(a) a y = f(x) b
Show that the two regions in Figure 4 have the same area. Then use the identity cos2 u = 1/2 (1 + cos 2u) to compute the second area. 1 y=√1-x² (A) 1 -X y = cos² u (B) + π/2 n.
How can we interpret the quantity I in Eq. (2) if a Eq (2) [ f(u(x))u'(x) dx = ru(b) f(u) du
Apply the method of Example 6 to ƒ(x) = cos x to determine ƒ'(π/5) accurately to four decimal places. Use a graph of ƒ to explain how the method works in this case. EXAMPLE 6 Estimate the
Find the derivative using the appropriate rule or combination of rules.y = (x3 + cos x)−4
Find all points on the graph of 3x2 + 4y2 + 3xy = 24 where the tangent line is horizontal (Figure 6). y X
Let A represent the area under the graph of y = x3 between x = 0 amd x = 1. In this problem, we will follow the process in Exercise 30 to approximate A.(a) As in (a)–(d) in Exercise 30, separately
Use Theorems 1–4 to show that the function is continuous.ƒ(x) = x1/3 cos 3x THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous
Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote. In Exercise 52, ƒ(x)
Find an equation of the tangent line at the point indicated.y = cos x, x = π/3
If dx/dt = 2 and y = x3, what is dy/dt when x = −4, 2, 6?
Consider a rectangular bathtub whose base is 18 ft2.At what rate is water pouring into the tub if the water level rises at a rate of 0.8 ft/min?
Evaluate (ƒ − g)(1) and (3 ƒ + 2g)'(1), assuming that ƒ'(1) = 3 and g'(1) = 5.
To which of the following does the Power Rule apply?(a) ƒ(x) = x2(c) ƒ(x) = xe(b) ƒ(x) = 2e(d) ƒ(x) = x−4/5
Find a and h such that ƒ(a + h) − ƒ(a)/h is equal to the slope of the secant line between (3, ƒ(3)) and (5, ƒ (5)).
Refer to the function ƒ whose graph is shown in Figure I.Estimate for h = 0.3. Is this difference quotient greater than or less than ƒ'(0.7)? 6- 5432 5- 0.5 1.0 1.5 2.0 -X
Which is the derivative of ƒ(5x)?(a) 5 ƒ'(x) (b) 5 ƒ'(5x) (c) ƒ'(5x)
Compute ƒ'(x) using the limit definition.ƒ(x) = x3
Figure 1 shows the graph of y4 + xy = x3 − x + 2. Find dy/dx at the two points on the graph with x-coordinate 0 and find an equation of the tangent line at each of those points. 2 -2. (1,
Let ƒ(x) = cos x. We can compute ƒ(n)(x) as follows: First, express n = 4m + r where m is a whole number and r = 0, 1, 2, or 3. Then determine ƒ(n)(x) from r. Explain how to do the latter step.
Compute the derivative. y = 8 1 + cot 0
Find the dimensions x and y of the rectangle of maximumarea that can be formed using 3 m of wire.(a) What is the constraint equation relating x and y?(b) Find a formula for the area in terms of x
Wire of length 12 m is divided into two pieces and each piece is bent into a square. How should this be done in order to minimize the sum of the areas of the two squares?(a) Express the sum of the
A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 12 km of fence available,
The rectangular bird sanctuary with one side along a straight river is to be constructed so that it contains 8 km2 of area. Find the dimensions of the rectangle to minimize the amount of fence
Find two positive real numbers such that the sum of the first number squared and the second number is 48 and their product is a maximum.
Find two positive real numbers such that they sum to 108 and the product of the first times the square of the second is a maximum.
A wire of length 12 m is divided into two pieces and the pieces are bent into a square and a circle. How should this be done in order to minimize the sum of their areas?
Find the positive number x such that the sum of x and its reciprocal is as small as possible. Does this problem require optimization over an open interval or a closed interval?
Find two positive real numbers such that they add to 40 and their product is as large as possible.
Find two positive real numbers x and y such that they add to 120 and x2y is as large as possible.
Find two positive real numbers x and y such that their product is 800 and x + 2y is as small as possible.
A flexible tube of length 4 m is bent into an L-shape. Where should the bend be made to minimize the distance between the two ends?

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