∫ calculus

1. The problem asks to find the derivative of $\cos(ax)$ with respect to $x$. 2. The formula for the derivative of $\cos(ax)$ is given as:

1. **Stating the problem:** We need to graph the curve defined by the equation $x=1+2y^2$ and consider its revolution about the $y$-axis between $y=1$ and $y=2$. 2. **Understanding

1. **Problem:** Find the derivative of $y = \frac{\sin x}{\cos x}$. 2. **Formula:** Use the quotient rule: if $y = \frac{u}{v}$, then

1. **State the problem:** We have the function $f(x) = 14 - x^2$.

1. **Problem Statement:** Find the x-values where the function $f$ is discontinuous based on the described piecewise graph. 2. **Understanding Discontinuity:** A function is discon

1. The problem asks to find all the $x$ values where the function $f$ is discontinuous based on the described piecewise graph. 2. Discontinuities occur where the function has jumps

1. The problem asks to find all the $x$ values where the function $f$ is discontinuous based on the described piecewise graph. 2. A function is discontinuous at points where there

1. The problem asks to find all the $x$ values where the function $f$ is discontinuous. 2. Discontinuities occur where the function has jumps, holes (open circles), or isolated poi

1. **Problem Statement:** Test the limit, continuity, and differentiability of the function $$f(x) = \begin{cases} \cos \frac{1}{x} & x \neq 0 \\ 0 & x = 0 \end{cases}$$

1. The problem asks for the expression of $x_{n+1}$ in Newton's Method to find the root of the function $$f(x) = \cos x (x^2 + 2x + 2).$$ 2. Newton's Method formula is:

1. **Problem statement:** Use Newton's method to calculate $x_4$ starting from $x_0=1$ for the function $$f(x) = \cos(x)(x^2 + 2x + 2).$$ 2. **Newton's method formula:**

1. The problem is to find the integral of $\sin(2x + y)$ with respect to $x$. 2. The integral formula for $\sin(ax + b)$ with respect to $x$ is:

1. The problem is to understand and apply the method of $u$-substitution in integration. 2. $u$-substitution is used to simplify integrals by substituting a part of the integral wi

1. **State the problem:** We need to evaluate the triple integral $$\int_0^{\pi/2} \int_0^y \int_0^x \cos(x + y + z) \, dz \, dx \, dy.$$\n\n2. **Understand the integral:** The int

1. **Problem statement:** Find and classify all stationary points of the function $$f(x) = x^3 - 6x^2 + 9x + 2$$. 2. **Find stationary points:** Stationary points occur where the f

1. **State the problem:** We need to find the equation of a curve such that at every point $(x,y)$ on the curve, the slope of the tangent line is equal to $-\frac{y}{x+y}$. 2. **Wr

1. **Problem statement:** Find the error of the 4th degree Taylor polynomial approximation for $\ln(\sec(0.4))$. 2. **Formula and explanation:** The Taylor series expansion of a fu

1. **Problem statement:** Find the first few coefficients $C_n$ of the Taylor series for $f(x) = \ln(\sec(x))$ centered at $a=0$, where the series is given by $\sum C_n x^n$. 2. **

1. **Problem Statement:** Given the function $$f(x) = \frac{x^2 - 2}{x^4}$$, find its first derivative $$f'(x)$$ and second derivative $$f''(x)$$. 2. **Rewrite the function:** To d

1. **Problem Statement:** Find the critical points, inflection points, intervals of increase/decrease, and concavity for the functions:

1. **Problem Statement:** Evaluate the integrals \(\int_0^{18} f(x) \, dx\), \(\int_0^{45} f(x) \, dx\), \(\int_{45}^{63} f(x) \, dx\), and \(\int_0^{81} f(x) \, dx\) by interpreti