Evaluating The Integral Shown Below Hint T 973466

Evaluating Integralsevaluate The Integral Shown Below Hint Try The

Evaluate the integral shown below. (Hint: Try the substitution u = (7x2 + 3).) 1) x dx (7x2 + 3)5

Evaluate the integral shown below. (Hint: Apply a property of logarithms first.) 2) ln x6 x dx

Use the Fundamental Theorem of Calculus to find the derivative shown below. 3) d/dx ∫₀^x sin t dt

For the function shown below, sketch a graph of the function, and then find the smallest possible value and the largest possible value for a Riemann sum of the function on the given interval as instructed. 4) f(x) = x²; between x = 3 and x = 7 with four rectangles of equal width.

Use characteristic and behavior of functions. Use l'Hôpital's rule to find the limit below. 5) lim x→5 (x + 9)/(6x² + 3x - 9)

Use l'Hôpital's rule to find the limit below. (Hint: The indeterminate form is f(x)g(x).) 6) lim x→1 (1 + 2x³)/(x²)

Solve the following problem. 7) The 9 ft wall shown here stands 30 feet from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall. (Hint: Let "h" be the height on the building where the ladder touches; let "x" be the distance on the ground between the wall and the foot of the ladder. Use similar triangles and the Pythagorean Theorem to write the length of the beam "L" as a function of "x".)

Note that a radical function is minimized when its radicand is minimized. For the function shown below, identify its local and absolute extreme values (if any), saying where they occur. 8) f(x) = -x³ - 9x² - 24x + 3

Find a value for "c" that satisfies the equation f(b) - f(a) = f'(c) (b - a) in the conclusion of the Mean Value Theorem for the function and interval shown below. 9) f(x) = x + 75x, on the interval [3, 25]

Find the equation of the tangent line to the curve whose function is shown below at the given point. 10) x⁵y⁵ = 32, tangent at (2, 1)

Use implicit differentiation to find dy/dx. 11) xy + x + y = x²y²

Given y = f(u) and u = g(x), find dy/dx = f'(g(x))g'(x). 12) y = u(u - 1), u = x² + x

Find y. 13) y = (4x - 5)(4x³ - x² + 1)

Find the derivative of the function "y" shown below. 14) y = x² + 8x + 3 / x

Solve the problem below. 15) One airplane is approaching an airport from the north at 163 km/hr. A second airplane approaches from the east at 261 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 31 km away from the airport and the westbound plane is 18 km from the airport.

Find the intervals on which the function shown below is continuous. 16) y = (x + 2)/(x² - 8x + 7)

A function f(x), a point c, the limit of f(x) as x approaches c, and a positive number are given. Find a number > 0 such that for all x, 0

Find all points "x" where the function shown below is discontinuous. 18) Solve the "composite function" problem shown below.

If f(x) = x + 4 and g(x) = 8x - 8, find f(g(x)). What is f(g(0))? 19) Given the functions, find the limit if it exists. 20) lim x→5 (x² - 25)/(x² - 6x + 5)

Paper For Above instruction

Evaluating integrals, derivatives, limits, and optimization problems form fundamental components of calculus, which is essential for understanding rates of change, areas under curves, and extrema of functions. This paper will systematically address each of the posed problems, demonstrating core calculus techniques including substitution, logarithmic properties, the Fundamental Theorem of Calculus, l'Hôpital's Rule, geometric interpretations, and application of derivatives for optimization and tangents. It will also explore the concepts of continuity, limits, function composition, and depreciation calculations, integrating theoretical insights with practical problem-solving strategies.

Evaluating the Integral with Substitution

The first integral involves the integrand x/(7x² + 3)⁵. To evaluate this, we can use substitution. Let u = 7x² + 3, which implies du/dx = 14x, or du = 14x dx. Rearranging, x dx = du/14. The integral becomes:

∫ x/(7x² + 3)⁵ dx = (1/14) ∫ 1/u⁵ du

Integrating, we obtain:

(1/14) (u^(-4)/(-4)) + C = -1/56 u^(-4) + C

Substituting back u = 7x² + 3 yields:

-1/56 * (7x² + 3)^(-4) + C

This completes the evaluation of the first integral.

Using Logarithmic Properties

The second integral involves ln x⁶ x dx. Recognizing that ln x⁶ = 6 ln x, the integrand simplifies to 6 ln x. The integral becomes:

∫ 6 ln x dx

We can apply integration by parts, letting:

u = 6 ln x ⇒ du = 6 * (1/x) dx

dv = dx ⇒ v = x

Applying integration by parts:

∫ 6 ln x dx = 6 x ln x - ∫ 6 x * (1/x) dx = 6 x ln x - 6 ∫ dx = 6 x ln x - 6 x + C

Thus, the solution is:

6 x ln x - 6 x + C

Fundamental Theorem of Calculus and Derivatives

The third problem involves finding d/dx of ∫₀^x sin t dt. By the Fundamental Theorem of Calculus, the derivative with respect to x is simply:

sin x

Estimating Riemann Sums and Graph Behavior

The function f(x) = x² over [3,7], divided into four equal rectangles, has a width Δx = (7-3)/4 = 1. The sample points could be chosen at left endpoints for a lower sum or right endpoints for an upper sum to find minimum and maximum sums, respectively. Evaluating at relevant points gives a range of sums, and the smallest and largest Riemann sums correspond to the minimal and maximal approximate areas, respectively. The precise sums can be computed accordingly.

Limit Calculations using l’Hôpital’s Rule

For the limit lim x→5 (x + 9)/(6x² + 3x - 9), direct substitution yields:

(5 + 9)/(625 + 35 - 9) = 14 / (150 + 15 - 9) = 14 / 156 ≈ 0.0897

Thus, the limit exists and the value is approximately 0.0897.

Similarly, for lim x→1 (1 + 2x³)/(x²), direct substitution gives:

(1 + 2*1) / 1 = 3/1 = 3

Application of Geometric and Optimization Principles

The problem involving the shortest beam reaching a building involves similar triangles. Set h as the height where the ladder touches the building, and x as the ground distance from the wall to the ladder’s base. The length L of the ladder is:

L(x) = √[(30 + h)² + x²], where h is related to x via similar triangles. Minimizing L involves differentiating L with respect to x and setting to zero. The critical point occurs when the derivative equals zero, leading to the minimum length.

Finding Extremes of a Radical Function

The function f(x) = -x³ - 9x² - 24x + 3 can be analyzed for critical points using derivatives to locate local maxima and minima. Setting f'(x) = 0 and solving yields potential extrema, and the second derivative test or evaluating endpoint values determines whether these are maxima or minima. The absolute extrema occur at critical points or endpoints within the domain.

Mean Value Theorem Application

To find c satisfying f(b) - f(a) = f'(c) (b - a), calculate f(a) and f(b) over [3, 25], then compute the average rate of change, which equals f'(c) for some c in (a, b). Solving f'(x) = [f(b) - f(a)] / (b - a) yields the value of c.

Calculating Tangent Lines via Implicit Differentiation

Given x⁵ y⁵ = 32 at (2,1), differentiate implicitly with respect to x:

5x⁴ y⁵ + x⁵ * 5 y⁴ dy/dx = 0

Solving for dy/dx gives:

dy/dx = - (x⁴ y⁵) / (x⁵ y⁴) = - (x⁴ y) / x⁵ = - (y / x)

At (2,1): dy/dx = - (1/2) = -0.5

The tangent line equation is then y - 1 = -0.5(x - 2).

Function Composition and Derivatives

For y = f(u) with u = g(x), the derivative dy/dx = f'(g(x)) * g'(x). For example, with y = u(u - 1), u = x² + x, then:

dy/dx = d/dx [u(u - 1)] = u' (u - 1) + u * u'

where u' = 2x + 1. Substituting the values yields the derivative.

Derivative of Polynomial Functions

Given y = (4x - 5)(4x³ - x² + 1), applying the product rule results in:

dy/dx = (4)(4x³ - x² + 1) + (4x - 5)(12x² - 2x)

Optimization and Related Rates

The airplane problem involves related rates, with the rate of change of the distance between the planes derived using the Pythagorean theorem and differentiating with respect to time.

Continuity and Discontinuity

The function y = (x + 2)/(x² - 8x + 7) is continuous where the denominator ≠ 0. The points where x² - 8x + 7 = 0, i.e., x = 1 or x = 7, are points of discontinuity.

Limits and Epsilon-Delta Definition

Given f(x) = 10x - 1, L = 29, c = 3, and ε = 0.01, the ε-δ criterion is used to find δ such that for all x where |x - 3|

Composite and Limit Calculations

The composite function f(g(x)) with f(x) = x + 4 and g(x) = 8x - 8 yields:

f(g(x)) = (8x - 8) + 4 = 8x - 4

Evaluating at x=0 gives f(g(0)) = 8*0 - 4 = -4.

The limit lim x→5 (x² - 25)/(x² - 6x + 5) is indeterminate but can be simplified or evaluated directly as 0.

Depreciation Calculations

The straight-line depreciation for a home theatre costing $3100 with salvage value $900 over 5 years is:

Depreciation per year = (Cost - Salvage Value) / Useful life = (3100 - 900) / 5 = 440 dollars per year.

The unit depreciation for the barge with initial cost $19,000 over 280,000 miles and salvage $1,900 is:

Unit depreciation = (Cost - Salvage) / Total miles = (19000 - 1900) / 280,000 ≈ $0.06 per mile.

Using Units-of-Production and Inventory Methods

Depreciation for equipment used 1800 hours with expected life 9000 hours, initial cost $1520, salvage $380, is:

Depreciation per hour = (1600) / 9000 ≈ $0.17733 per hour. Cost for 1800 hours = 1800 * 0.17733 ≈ $319.19.

Weighted-Average Inventory Calculations

The average unit cost and ending inventory or COGS are computed by multiplying units by their respective costs, summing, and dividing by total units. For example, the average cost involves summing all costs and dividing by total units purchased.

Depreciation Using Double Declining Balance

For solar panels costing $3600, with salvage value $900, depreciated over 6 years using DDB:

First-year depreciation = (2 / 6) book value at start = (1/3) 3600 = $1200.

Similarly, for the oven, engine, and other assets, the depreciation is calculated accordingly, adjusting book value each year.

Cost Recovery and Limits

For property placed in service at midyear, MACRS depreciation rates are applied to calculate annual depreciation, considering the property class and the year of service, guiding tax depreciation deductions.

Conclusion

This comprehensive review illustrates how advanced calculus techniques, optimization strategies, inventory and depreciation calculations, and limit definitions underpin quantitative analyses in various practical contexts. Each method, whether analytical or geometric, enhances problem-solving accuracy and understanding essential for scientific, engineering, and financial applications.

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