∫ calculus

1. **Problem 29:** Find the derivative of \(y = (x^2 + 2)(x^4 + 4)^4\) using logarithmic differentiation. 2. Take the natural logarithm of both sides:

1. **Problem 1: Find the instantaneous velocity and acceleration of the drone at $t=5$ given $s(t) = 7t^3 - 3t^2 - 5t + 2$.** 2. The instantaneous velocity is the first derivative

1. **Problem (I):** Calculate $$\lim_{x \to 0} \frac{27 - (3 + x)^3}{x}$$ 2. **Step 1:** Expand the cube in the numerator:

1. Problem (a): Find $$\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{2} + h\right) - 1}{h}$$. 2. Use the identity $$\sin\left(\frac{\pi}{2} + h\right) = \cos h$$.

1. **Problem 1:** Find the tangent line to $y=f(x)=\sqrt{x}+1$ at $a=3$. 2. First, compute $f(3)$:

1. The problem asks to determine the continuity of the function $g(x)$ at specific points based on the graph description. 2. At $x = -2$, the graph has a closed dot at $(-2,0)$ and

1. The problem describes the continuity and discontinuity of the function $g$ at specific points on the interval $[-2,3]$ based on the graph. 2. At $x = -2$, $g$ is continuous from

1. **State the problem:** We want to determine whether the series $$\sum_{n=2}^{\infty} \frac{n^2}{n^3 + 1}$$ converges or diverges using the integral test. 2. **Set up the integra

1. **State the problem:** Determine whether the series $$\sum_{n=1}^{\infty} \left(3^n + 1 \cdot 4^{-n}\right)$$ converges or diverges, and if it converges, find its sum. 2. **Rewr

1. **State the problem:** We need to evaluate the integral $$\int e^x \sqrt{81 - e^{2x}} \, dx.$$\n\n2. **Substitution:** Let $$u = e^x.$$ Then, $$du = e^x dx = u dx \implies dx =

1. **State the problem:** We want to show that the sum of the areas of the upper approximating rectangles under the curve $f(x) = 5x^2$ on the interval $[0,2]$ approaches $\frac{40

1. **Problem a:** Find an appropriate trigonometric substitution for $$\int (5x^2 - 3)^{3/2} \, dx$$ given $$x = \sqrt{\frac{3}{5}} \sec \theta$$. Step 1: Recognize the form inside

1. **State the problem:** We need to evaluate the integral $$\int 5 \sin^4(x) \cos^2(x) \, dx.$$\n\n2. **Rewrite the integral:** Express powers of sine and cosine in terms of power

1. **State the problem:** We need to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$x^3 + y^3 = 5.$$\n\n2. **Differentiate both sides with respect to $x$:**\nU

1. **State the problem:** We need to sketch a function $f$ continuous on $[1,5]$ with a second derivative on $(1,5)$ satisfying:

1. **State the problem:** We want to find the integral $$\int \frac{4x - 1}{(x + 1)(x + 2)} \, dx.$$\n\n2. **Use partial fraction decomposition:** Express the integrand as $$\frac{

1. The problem asks to find the points of inflection of $f$ and the intervals where $f$ is concave down, given the graph of its derivative $f'$. 2. Points of inflection of $f$ occu

1. The problem is to find the limit as $x \to \infty$ of the expression $$\frac{t^4 + 2x^5 - 3x^5}{x^5}$$. 2. First, divide each term in the numerator by the highest power of $x$ i

1. Problem (a): Differentiate $$y = \frac{x^3 + 3x}{(x+1)(x+2)}$$ 2. First, simplify the denominator:

1. **State the problem:** We have the curve defined by $$y = 4\sqrt{x} - \frac{x}{2} + 1$$ for $$0 \leq x \leq 64$$. We need to find values of $$a, b, c$$ at $$x=16, 32, 48$$ respe

1. **State the problem:** We are given the rate of change of cost per hour $P$ with respect to time $t$ as $$\frac{dP}{dt} = 20 - \frac{980}{t^2}, \quad 0 < t \leq 12.$$ We need to