Evaluate The Integral Shown Below — Hint: Try The Substituti
evaluate The Integral Shown Below Hint Try The Substitution U
Evaluate the integral shown below. (Hint: Try the substitution u = (7x² + 3).)
Evaluate the integral using the substitution u = 7x² + 3. First, differentiate u with respect to x: du/dx = 14x, which implies du = 14x dx. To proceed, rewrite the integral in terms of u. Express dx in terms of du and x, then substitute into the original integral and integrate with respect to u. Finally, back-substitute to express the answer in terms of x.
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The integral problem prompting substitution u = 7x² + 3 is a classic example of calculus techniques involving substitution to simplify integration. In such cases, recognizing the inner function and its differential allows for a smoother integration process. Let us delve into the step-by-step solution.
The original integral is assumed to be of the form ∫f(x) dx, where f(x) involves 7x² + 3, possibly in the numerator or denominator. By setting u = 7x² + 3, the differential du becomes 14x dx, which neatly relates to the existing x dx term in the integral. Rearranged, du/14x = dx. This substitution transforms the original integral into an integral in terms of u, often simplifying the integrand to a basic polynomial or rational function (depending on the original integral).
After substitution, the integral becomes ∫(some function of u) du/ (14x). Here, it's essential to express x in terms of u: x = √[(u - 3)/7], which may be necessary depending on the integral's form. Proceeding with substitutions, the integral simplifies significantly, allowing for direct integration of u-based functions. After integrating, substitute back u = 7x² + 3 to get the solution expressed in terms of x.
This method illustrates the power of substitution techniques in calculus. Recognizing the inner functions within integrals enables teachers and students alike to tackle complex integrals with confidence and precision, reinforcing core calculus principles.
evaluate The Integral Shown Below Hint Try The Substitution U
Evaluate the integral shown below. (Hint: Apply a property of logarithms first.)
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The instruction suggests applying a property of logarithms before evaluating the integral. This approach indicates that the integrand likely involves logarithmic expressions, making properties like the product rule, quotient rule, or power rule of logs essential prior to integration.
For example, if the integrand is of the form ∫ln(x) dx, or involves products or quotients such as ∫(ln(x))/x dx, then properties like ln(ab) = ln a + ln b, or ln(a/b) = ln a - ln b, help simplify the integrand for easier integration.
Applying a property of logarithms might involve rewriting the integrand to isolate a function whose integral is known or more straightforward, such as rewriting a product as a sum of logs or a power as a multiple of a log. Once simplified, standard integral formulas, such as ∫ln(x) dx = x ln x - x + C, can be employed.
For instance, if the integrand involves a composite logarithmic expression, rewriting it using properties may reveal a pattern that directly matches a known derivative, simplifying integration. Recognizing these properties is crucial in reducing complex logarithmic integrals to manageable forms, demonstrating the importance of algebraic manipulation before integration in calculus.
evaluate The Integral Shown Below Hint Try The Substitution U
Use the Fundamental Theorem of Calculus to find the derivative shown below.
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The instruction directs to apply the Fundamental Theorem of Calculus (FTC) to find a derivative associated with a function defined by an integral. The FTC states that if F(x) = ∫a^x f(t) dt, then F'(x) = f(x). This theorem bridges differentiation and integration, enabling us to find derivatives of integrals with variable upper limits directly.
Suppose the function given is of the form F(x) = ∫a^x f(t) dt, where f(t) is continuous on [a, x]. By applying the FTC, the derivative F'(x) simplifies immediately to f(x). This process often involves verifying the conditions of the theorem, ensuring the integrand is continuous, and then directly substituting the upper limit x into the integrand to find the derivative.
This approach simplifies complex differentiation tasks, turning integral expressions into straightforward evaluations of the integrand at x. Understanding and applying the FTC correctly is fundamental to solving various problems involving variable limits, enabling efficient computation in calculus.
Use The Fundamental Theorem of Calculus To Find The Derivative
For the given function, sketch a graph and then find the smallest and largest possible values for a Riemann sum on the specified interval.
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Calculating a Riemann sum involves partitioning the interval into subintervals, computing the function's value at specified points within each subinterval, and summing these values multiplied by the subinterval widths. The smallest Riemann sum corresponds to choosing points where the function attains its minimum on each subinterval, while the largest corresponds to its maximum. Graphing the function helps visualize these extrema within each subinterval.
Sketching the function provides insight into where the function reaches local minima and maxima, guiding the selection of sample points for Riemann sums. The smallest sum is often approximated by left endpoints or points at which the function is minimal in each subinterval, and vice versa for the largest sum.
As the partition gets finer (more subintervals), these sums converge to the definite integral, thanks to the properties of Riemann sums and the function’s continuity. This process exemplifies the relationship between the geometric interpretation of integrals and their numerical approximations, emphasizing the importance of graphing in understanding and estimating integral values.
Use L’Hopital’s Rule To Find The Limit
lim x→∞ (5x + 9) / (6x² + 3x - 9). Use L’Hôpital’s Rule to evaluate this limit.
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The problem involves evaluating the limit of a rational function as x approaches infinity, which results in an indeterminate form of ∞/∞. L'Hôpital’s Rule states that if the limit results in an indeterminate form, then the limit of the quotient is equal to the limit of the derivatives of numerator and denominator, provided the latter limit exists.
Applying L'Hôpital’s Rule involves differentiating numerator and denominator separately: d/dx (5x + 9) = 5, and d/dx (6x² + 3x - 9) = 12x + 3. The new limit becomes lim x→∞ 5 / (12x + 3). As x approaches infinity, the denominator approaches infinity, so the limit is 0.
This application showcases how L’Hôpital’s Rule simplifies limits of indeterminate forms involving infinity, emphasizing the importance of derivative rules in analyzing the behavior of functions at extremes in calculus.
Use L’Hopital’s Rule To Find The Limit
lim x→∞ (5x + 9) / (6x² + 3x - 9). The indeterminate form is f(x)g(x).
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The instruction indicates evaluating a limit involving a product of functions as x approaches infinity, which might be in an indeterminate form such as 0×∞, ∞×0, or similar. L’Hôpital’s Rule can be extended to such cases by rewriting the product as a quotient—typically, by expressing, for example, as f(x)/[1/g(x)] or vice versa. This enables application of the rule.
In the given case, the product form suggests examining the functions involved, rewriting the expression so that the indeterminate form is explicitly a quotient. Then, differentiating numerator and denominator separately allows for the evaluation of the new limit. Alternatively, recognizing the dominant terms, the limit can be directly approximated, but using L’Hôpital’s Rule ensures a rigorous approach, especially when the behavior is less obvious.
This technique demonstrates the flexibility of L’Hôpital’s Rule beyond simple quotient forms, applying it skillfully to analyze limits involving products in advanced calculus contexts.
solve the following problem
The 9 ft wall stands 30 feet from the building. Find the length of the shortest straight beam that will reach the side of the building from the ground outside the wall.
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This problem involves optimization in geometry, specifically finding the shortest length of a beam that reaches from a point outside a wall to the side of a building. It can be modeled using right triangles and calculus principles. The wall height and distance, along with the position of the beam, serve as variables in minimizing the length of the beam.
By defining variables for the segment outside the wall, applying the Pythagorean theorem, and setting up an expression for the beam's length in terms of these variables, the problem reduces to an optimization problem. Differentiating this expression with respect to the variable and finding critical points allows determination of the minimum length. The solution often involves calculus deductions about the function's behavior and the use of derivatives to locate the minimal configuration.
This problem illustrates an application of calculus in real-world scenarios, emphasizing the importance of geometric reasoning combined with differentiation to optimize structural elements.
find a value for “c” that satisfies the equation f b( ) - f a( ) / (b - a) = f’(c)
This is the Mean Value Theorem: find c in the given interval such that the instantaneous rate of change at c equals the average rate of change over the interval.
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The Mean Value Theorem (MVT) states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that f’(c) = [f(b) - f(a)] / (b - a). To find this c, you first calculate the average rate of change over [a, b], then differentiate the function to find f’, and finally solve the equation f’(c) = (f(b) - f(a)) / (b - a) for c.
This process demands understanding the function's behavior across the interval and the application of differentiation to locate the precise point where the instantaneous rate matches the average. It is a fundamental theorem linking average and instantaneous rates of change in calculus.
find the equation of the tangent line to the curve at a point
The curve is given implicitly by x^5 y^5 = 32, and the tangent is at (2, ?). Use implicit differentiation to find dy/dx.
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To find the tangent line to the curve x^5 y^5 = 32 at (2, y), first find the corresponding y-value by substituting x=2: 2^5 y^5 = 32. Simplify to obtain y^5 = 32/32 = 1, so y = 1.
Next, differentiate both sides implicitly with respect to x: d/dx [x^5 y^5] = 0. Applying the product rule, we get:
5x^4 y^5 + x^5 * 5y^4 dy/dx = 0.
Factor out common terms:
5 x^4 y^4 ( y + x dy/dx )= 0.
Divide both sides by 5 x^4 y^4 (assuming x ≠ 0 and y ≠ 0):
y + x dy/dx = 0.
Solve for dy/dx:
dy/dx = - y / x.
At the point (2, 1), substitute x=2 and y=1:
dy/dx = -1/2.
Finally, write the equation of the tangent line with point-slope form:
y - 1 = (-1/2)(x - 2).
Simplify to get the equation of the tangent line:
y = (-1/2)(x - 2) + 1.
given y = f ( u ) and u = g ( x ), find dy/dx
Given the chain rule y = f(g(x)), find dy/dx = f ’(g(x)) * g’(x).
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The chain rule states that if y = f(g(x)), then the derivative dy/dx is the product of the derivative of the outer function evaluated at the inner function, f ’(g(x)), multiplied by the derivative of the inner function, g’(x). To compute dy/dx, first differentiate f with respect to its argument u = g(x), then multiply by g’(x). This process efficiently encapsulates the composite nature of the functions.
find y’
This appears to be a prompt to compute the derivative of a specified function y. The method involves applying differentiation rules suitable to the form of y, such as the power rule, product rule, quotient rule, or chain rule, depending on y’s expression.
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Without the explicit form of y, a general approach involves identifying the type of function (polynomial, rational, composite) and applying the appropriate differentiation rule accordingly. For polynomials, use power rules; for products, the product rule; for quotients, the quotient rule; and for composite functions, the chain rule. The process entails differentiating step-wise to obtain an expression for y’. If y is given explicitly, directly differentiate it to find y’.
Find the derivative of the function “y”
The instruction indicates calculating y’, the derivative of a specific function y. The approach depends on the form of y and involves applying relevant differentiation rules.
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Similar to the previous problem, deriving y’ requires the explicit form of y. Once the expression is known, differentiate term by term, applying the power rule, product rule, quotient rule, or chain rule as needed. Careful stepwise differentiation ensures accuracy. Present the final expression as the derivative y’ with respect to the independent variable.
solve the problem of airplane approach
One airplane approaches from the north at 163 km/hr; another 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 from the airport and the westbound plane is 18 km away from the airport. The velocities are given as dy/dt = -163 km/hr and dx/dt = -261 km/hr, corresponding to southward and westward approaches. The distance d between the planes is d = x² + y², where x and y are distances from the airport.
Differentiate d = x² + y² with respect to time t: dd/dt = 2x dx/dt + 2y dy/dt. Substitute the given values and compute the rate of change in the distance between the planes at the specified distances, providing the exact numerical result.
find the intervals on which the function shown below is continuous
Given the function y = (x + 2) / (x² - 8x + 7), analyze the points where the function is continuous. Discontinuities occur at points where the denominator is zero; thus, solve x² - 8x + 7 = 0 to find the critical points. The function is continuous everywhere except where the denominator equals zero, which corresponds to these roots. Therefore, the interval of continuity is the real line minus these points.
find delta for epsilon-delta proof
Given a function f(x), a point c, a limit L, and a positive epsilon, find delta > 0 such that for all x with 0
solve composite function problem
Given functions f(x) = x + 4 and g(x) = 8x - 8, find (f ◦ g)(x) = f(g(x)).
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The composition (f ◦ g)(x) means applying g(x) first, then f to the result. Calculate g(x): g(x) = 8x - 8. Then substitute into f: f(g(x)) = g(x) + 4. Since g(x) = 8x - 8, the composite function is:
f(g(x)) = (8x - 8) + 4 = 8x - 4.
Evaluate at x = 0: f(g(0)) = 8(0) - 4 = -4.
This process exemplifies the mechanics of function composition, emphasizing the sequential application of functions and their simplified forms.
Find the limit as x approaches 5 of (x^2 - 25) / (x^2 - 6x + 5)
Factor numerator and denominator: numerator: (x - 5)(x + 5); denominator: (x - 1)(x - 5). Cancel the common factor (x - 5):
Limit becomes lim x→5 (x + 5) / (x -