∫ calculus

1. **State the problem:** Differentiate the functions given using the power rule for derivatives. 2. **Recall the power rule:** If $$f(x) = ax^n$$ where $$n \in \mathbb{N}$$ and $$

1. **Problem 2a:** Find the gradient of the secant line $PQ$ where $P(4,4)$ and $Q(4+h, (4+h)^2 - 3(4+h))$ lie on $y = f(x) = x^2 - 3x$. 2. The gradient of the secant $PQ$ is given

1. **State the problem:** We want to find the gradient of the tangent to the curve $y = x^3$ at the point $(2,8)$ by first calculating the gradient of the secant line through point

1. **State the problem:** Rewrite the integral $$\int \frac{\sqrt{16x^2 - 81}}{x^3} \, dx$$ using the substitution $$x = \frac{9}{4} \sin \theta$$.

1. **State the problem:** Rewrite the integral $$\int \frac{\sqrt{16x^2 - 81}}{x^3} \, dx$$ using the substitution $$x = \frac{9}{4} \sin \theta$$.

1. **State the problem:** Differentiate the function $$y=\ln(2+\sqrt{x})^2$$ with respect to $x$. 2. **Rewrite the function:** Using the property of logarithms, $$\ln(a)^2 = (\ln a

1. **State the problem:** Differentiate the function $$y=(2+\sqrt{x})^2$$ with respect to $$x$$. 2. **Recall the formula:** Use the chain rule for differentiation. If $$y = [u(x)]^

1. **State the problem:** We are given the function $$f(x) = x^3 - 7x^2$$ and need to find its derivative $$f'(x)$$, solve $$f'(x) = 0$$, and find the points on the graph where the

1. **State the problem:** We are given the function $$f(x) = \frac{x^3 + 7x^2 + x - 5}{x}$$ and asked to rewrite it by dividing each term in the numerator by the denominator, then

1. **State the problem:** We are given graphs of functions $f$ for Problems 29 to 32. For each, we need to find: (a) The $x$-values where $f$ has local maxima and the corresponding

1. **State the problem:** Calculate the integral $$\int_{-\infty}^{\infty} e^{-x^2} \, dx$$. 2. **Formula and important rules:** This is a Gaussian integral, a well-known integral

1. **Problem Statement:** Determine the increasing and decreasing intervals for the graphs labeled #23 and #24 based on the given information. 2. **Given Information:**

1. **Problem Statement:** We are given graphs of functions $f$ for problems 30, 31, and 32. We need to find the local maxima and minima points for each function.

1. **Stating the problem:** We want to evaluate the integral $$\int \frac{x^2(1+\sin(x)\cos(x))^2}{(x^2+1)^3(\sin(x)+\cos(x))^4 \ln(x^2+1)} \, dx$$

1. **State the problem:** We need to evaluate the integral $$\int x^2(1+\sin x \cos x)^2 (x^2+1)^3 (\sin x + \cos x)^4 \ln(x^2+1) \, dx.$$\n\n2. **Analyze the integral:** This inte

1. **State the problem:** We need to find the limit $$\lim_{x \to 4} \frac{x-4}{(x-4)(\sqrt{x}+2)}$$. 2. **Identify the issue:** Direct substitution of $x=4$ gives $$\frac{4-4}{(4-

1. **State the problem:** Find the limit

**Problem:** Evaluate $\lim_{x\to\infty}\left(\frac{4^x+9^x}{x}\right)^{1/x}$. 1. Use the dominant-term rule: when $x\to\infty$, the larger exponential base dominates, so factor ou

1. **State the problem:** We are given a function $y = f(x)$ with discontinuities at $x = -1$, $x = 0$, and $x = 1$. We need to find these discontinuities, explain why the formal d

1. **Problem:** Prove that $$\lim_{x \to 0} x^4 \cos\left(\frac{2}{x}\right) = 0$$. 2. **Formula and rules:** We use the Squeeze Theorem. Since $$-1 \leq \cos\left(\frac{2}{x}\righ

1. **State the problem:** We want to find the volume of a solid whose base is a right triangle with vertices at $(0,0)$, $(7,0)$, and $(0,7)$.