Calculus Limits, Derivatives, and Integrals

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 $\frac{t - \sqrt{3 + 4}}{4 - t}$.

What is the third derivative $\frac{d^4y}{dx^4}$ when given $y(x) = x^4 - 4x^3 + 3x^2 - 5x$?

Evaluate the limit as $x$ approaches infinity for the expression $lim_{x→∞} (2x^4−x^2+8x−5x^4+7)$.

What is the result of differentiating $f(x) = (ax^3 + bx)$ with respect to $x$?

Evaluate the limit as $x$ approaches negative infinity for the expression $lim_{x→−∞} (x^2−5t−9)/(2x^4+3x^3)$.

Given $y(x) = x^4 - 4x^3 + 3x^2 - 5x$, what is the evaluated expression for $4y$ with respect to $x$?

What is the result of integrating the expression $∫(x^3 + 3x^2 + 2x + 4)$?

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 - 18h evaluates 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 + 35 using the first principle results in 4x - 16.
Derivative of f(x) = (ax³ + bx) is 3ax² + 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) gives y' = 5x³ + 4x³.

Integral Calculations

Integral of the polynomial function ∫(x³ + 3x² + 2x + 4)dx yields x⁴/4 + x³ + x² + 4x + c.
Evaluate ∫e^(4x)dx, which evaluates to e^(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 be 4πr³/3.

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.

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!