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Questions and Answers
What is the limit as $x$ approaches $-\infty$ for the expression $\frac{x^2 - 5t - 9}{2x^4 + 3x^3}$?
Which derivative represents $f(x) = 2x^2 - 16x + 35$ using the first principle?
What is the derivative of $y = 3\sqrt{x^2}(2x - x^2)$ with respect to $x$?
Evaluate the limit as $t$ approaches 4 for $\frac{t - \sqrt{3 + 4}}{4 - t}$.
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What is the third derivative $\frac{d^4y}{dx^4}$ when given $y(x) = x^4 - 4x^3 + 3x^2 - 5x$?
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Evaluate the limit as $x$ approaches infinity for the expression $lim_{x→∞} (2x^4−x^2+8x−5x^4+7)$.
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What is the result of differentiating $f(x) = (ax^3 + bx)$ with respect to $x$?
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Evaluate the limit as $x$ approaches negative infinity for the expression $lim_{x→−∞} (x^2−5t−9)/(2x^4+3x^3)$.
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Given $y(x) = x^4 - 4x^3 + 3x^2 - 5x$, what is the evaluated expression for $4y$ with respect to $x$?
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What is the result of integrating the expression $∫(x^3 + 3x^2 + 2x + 4)$?
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Study Notes
Limit Evaluations
- Limit as x approaches negative infinity for
(x² - 5t - 9) / (2x⁴ + 3x³)evaluates to 4. - Limit as x approaches infinity for
(6e^(4x) - e^(-2x)) / (8e^(4x) - e^(2x) + 3e^(-x))evaluates to 3/4. - Limit as h approaches zero for
(−3 + h)² - 18hevaluates to 12. - Limit as t approaches 4 for
(t - √(3 + 4)) / (4 - t)evaluates to -3/8.
Derivative Calculations
- Derivative of the function
f(x) = 2x² - 16x + 35using the first principle results in4x - 16. - Derivative of
f(x) = (ax³ + bx)is3ax² + b. - Given
y(x) = x⁴ - 4x³ + 3x² - 5x, the fourth derivative evaluated yields 24. - Explore differentiation with respect to a function involving roots and products, e.g.,
y = 3√(x²)(2x - x²)givesy' = 5x³ + 4x³.
Integral Calculations
- Integral of the polynomial function
∫(x³ + 3x² + 2x + 4)dxyieldsx⁴/4 + x³ + x² + 4x + c. - Evaluate
∫e^(4x)dx, which evaluates toe^(4x)/4 + c. - Find the integral of
sec³(x) tan(x)dx, results show it leads to some trigonometric transformations. - The volume of a sphere derived from the semicircle
y = √(r² - x²)revolving about the x-axis is calculated to be4πr³/3.
Additional Limit Evaluations
- Limit as x approaches infinity for
(2x⁴ - x² + 8x - 5x⁴ + 7)evaluates to 1/2. - Confirm limits and derivatives repeatedly to ensure proficiency.
Practice Application
- Several questions and problems emphasize key calculus concepts including limits, differentiation, and integration in straightforward manners.
- Use provided answers to self-assess understanding and problem-solving capabilities related to calculus.
Conclusion
- Solidify understanding and computation skills in central calculus topics through repeated practice and evaluation of limits, derivatives, and integrative functions.
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Description
This quiz assesses your understanding of limits, derivatives, and integrals in calculus. You'll tackle problems involving limits as x approaches infinity, derivative calculations using the first principle, and integral evaluations of polynomial functions. Test your calculus skills and see how well you can apply these concepts!
