### A hurricane wind blows across a $6.00 \,m\times 15.0\, m$ flat roof at a speed of $130\, km/h$. Is the air pressure above the roof higher or lower than the pressure inside the house? Explain.

What is the pressure difference? How much force is exerted on the roof? If the roof cannot withstand this much force, will it “blow in” or “blow out?” The main objective of this problem is to determine the air pressure, pressure difference, and the force exerted by hurricane wind on the roof. Bernoulli’s equation is being […]

### If xy+8e^y=8e , find the value of y” at the point where x=0.

This question aims to find the value of the second derivative of the given non-linear equation.  Nonlinear equations are those which show up as curved lines when graphed. The degree of such an equation is two or more, but not less than two. The curvature of the graph increases as the value of the degree […]

### If xy+6e^y=6e, find the value of y” at the point where x=0.

This question aims to find the second derivative of the given implicit function. A function’s derivatives describe the rate of change of that function at a given point. If the dependent variable, say $y$, is a function of the independent variable, say $x$, we usually express $y$ in terms of $x$. When this occurs, $y$ […]

### Each limit represents the derivative of some function f at some number a.

Find the number $a$ and the function $f$ given the following limit: [lim_{tto 1} frac{t^4 + t – 2}{t-1}] The aim of this question is to learn the differentiation (calculation of derivative) from first principles (also called by definition or by ab-initio method). To solve this question, one needs to know the basic definition of […]

### Find transient terms in this general solution to a differential equation, if there are any

$y=(x+C)(dfrac{x+2}{x-2})$ This article aims to find the transient terms from the general solution of the differential equation. In mathematics, a differential equation is defined as an equation that relates one or more unknown functions and their derivatives. In applications, functions generally represent physical quantities, derivatives represent their rates of change, and a differential equation defines the relationship between them. Such […]

### Solve differential equation ty’+(t+1)y=t , y(ln2)=1 , t>0

In this question, we have to find the Integration of the given function $t y^prime + ( t + 1) y = t$ by using different integration rules. The basic concept behind this question is the knowledge of derivatives, integration, and the rules such as the product and quotient integration rules. Expert Answer […]

### Find the directional derivative of f at the given point in the direction indicated by the angle θ.

This question aims to find the directional derivative of the function f at the given point in the direction indicated by the angle $theta$. A directional Derivative is a type of derivative that tells us the change of the function at a point with time in the vector direction. We also find partial derivatives according […]

### Population y grows according to the equation dy/dt = ky, where k is a constant and t is measured in years. If the population doubles every ten years, then the value of k is?

This problem aims to familiarize us with the law of natural growth and decay. The concept behind this problem is exponential growth formulas and their derivates. We have seen that numerous entities grow or decay according to their size. For instance, a group of viruses may triple every hour. After some time $(t)$, if the […]

### Given a standard normal distribution, find the area under the curve that lies (a) to the left of z=-1.39; (b) to the right of z=1.96 ; (c) between z=-2.16 and z = -0.65; (d) to the left of z=1.43 ; (e) to the right of z=-0.89; (f) between z=-0.48 and z= 1.74.

This article aims to find the area under the curve for a standard normal distribution. A normal probability table is used to find the area under the curve. The formula for the probability density function is: [ f ( x ) = dfrac{ 1 }{ sigma sqrt 2 pi } e ^ {-dfrac{ 1 }{ 2 } ( […]

### At what point does the curve have maximum curvature? y = 7 ln(x)

The aim of this question is to introduce the local maxima and minima of a curve. Local maxima are defined as the point where the absolute value of the function is maximum. Local minima are defined as the point where the absolute value of the function is minimum. To evaluate these values, we need to […]

### Show that the equation has exactly one Real root 2x+cosx=0.

This question aims to find the real root of the given equation using the Intermediate theorem and Rolle’s theorem. If the function is continuous on the interval [c,d] then there should be an x-value in the interval for every y-value that lies in the f(a) and f(b). The graph of this function is a curve […]

### Determine a region whose area is equal to the given limit. Do not evaluate the limit.

[lim_{ntoinfty}sum_{i=1}^{n}frac{pi}{4n}{tanleft(frac{ipi}{4n}right)} ] The purpose of this article is to find the region having an area under the curve that is represented by a given limit. The basic concept behind this guide is the use of the Limit Function to determine an area of the region. The area of a region that covered the space above […]

### If f(2)=10 and f'(x)=x^2f(x) for all x, find f”(2).

The aim of this question is to learn how to evaluate the values of a higher order derivative without explicitly declaring the function itself.   To solve such problems, we may need to solve the basic rules of finding the derivatives. These include the power rule and product rule etc. According to the power rule […]

### Express the plane z=x in cylindrical and spherical coordinates.

This question aims to find the cylindrical and spherical coordinates of the plane z = x. This question is based on the concept of coordinate systems from calculus. Cylindrical and spherical coordinate systems are expressed in the cartesian coordinate systems. A spherical object like a sphere of a ball is best expressed in a spherical […]

### Find the exponential function f(x) = a^x whose graph is given.

This problem aims to find the exponential function of a given curve, and there lies a point on that curve at which the solution will proceed. To better understand the problem, you need to have good knowledge of exponential functions and their decay and growth rate techniques. First, let’s discuss what an exponential function is. […]

### Coffee is draining from a conical filter into a cylindrical coffee pot of radius 4 inches at the rate of 20 cubic inches per minute. How fast is the level in the pot rising when the coffee in the cone is 5 inches deep. How fast is the level in the cone falling then?

The aim of this question is to use the geometric formulas of volume of different shapes for the solution of word problems. The volume of the cone-shaped body is given by: [ V = dfrac{ 1 }{ 3 } pi r^2 h ] Where h is the depth of the cone. The volume of the cylindrical-shaped […]

### Determine whether f is a function from Z to R for given functions

$f(n) =pm n$ $f(n) = sqrt {n^2 + 1}$ $f(n) = dfrac{1}{n^2 -4}$ The aim of this question is to find out if the given equations are functions from Z to R. The basic concept behind solving this problem is to have sound knowledge of all sets and the conditions for which a given equation is […]

### Find the surface area of the torus shown below, with radii r and R.

The main objective of this question is to find the surface area of the given torus with the radii represented by r and R. This question uses the concept of the torus. A torus is basically the surface revolution generated as a result of rotating the circle in the three-dimensional space. Expert Answer In this […]

### For what value of the constant c is the function f continuous on (-∞, ∞)?

– Given Function [ fleft( xright)= bigg{begin{array}{rcl} cx^2+2x, & x<2 \ x^3-cx, & x≥2 end{array}] The aim of the question is to find the value of constant c for which the given function will be continuous on the whole real number line. The basic concept behind this question is the concept of Continuous Function. A function […]

### Find the first partial derivatives of the function f(x, y) = (ax + by)/(cx + dy)

The aim of this question is to find the first-order partial derivatives of an implicit function made up of two independent variables. The basis for this solution resolves around the quotient rule of derivatives. It states that if $u$ and $v$ are two functions, then the derivative of the quotient $frac{u}{v}$ can be calculated using […]

### Use definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.

$f ( x ) = dfrac { 2 x } { x ^ { 2 } + 1 } , 1 leq x leq 3$ This article aims to write the expression for the area under the graph. The article uses the concept of definition $2$ to find the expression for the area under the […]

### Determine whether the sequence converges or diverges. If it converges, find the limit.

$a _ { n } = dfrac { n ^ { 4 } } { n ^ { 3 } – 2 n }$ This article aims to determine whether the sequence converges or diverges. The article uses the concept to determine whether the sequence is convergent or divergent. When we say that a […]

### Show that if A^2 is the zero matrix, then the only eigenvalue of A is 0.

The aim of this question is to prove the statement for only the eigenvalue of $A$ to be zero. The concept behind this question is the knowledge of eigenspace and eigenvalue. Expert Answer Suppose that a non-zero value $lambda$ is an eigenvalue of the vector $A$ and the corresponding eigenvector = $vec{ x }$. […]

### Suppose that f(5)=1, f'(5)=6, g(5)=-3, and g'(5)=2. Find the following values of (fg)'(5), (f/g)'(5), and (g/f)'(5).

This problem aims to familiarize us with different methods to solve a differential. The concept required to cater to this problem mostly relates to ordinary differential equations. We define an ordinary differential equation or most commonly known as ODE, as an equation that has one or additional functions of a single independent variable given with […]

### Two runners start a race at the same time and finish in a tie.

The main objective of this question is to prove that the two runners have the same speed during some interval of time in the race. This question uses the concept of Calculus and Rolle’s theorem. In Rolle’s theorem, two conditions must be satisfied by a function that is defined in the interval [a,b]. The two conditions […]

### Find the point on the line y=5x+3 that is closest to the origin.

This question aims to find a point that is nearest to the origin and that is lying on the given line $y$ = $5x$ + $3$. The distance formula is used to calculate the distance between two sets of points where ( $x_1$ , $y_1$ ) is the first set of points and ( $y_1$ , $y_2$ […]

### Find the constant a such that the function is continuous on the entire real line.

Given Function: [ fleft( xright)= bigg{begin{array}{rcl} x^3, & x≤2 \ ax^2, & x>2 end{array}] The aim of the question is to find the value of constant a for which the given function will be continuous on the whole real number line. The basic concept behind this question is the knowledge of the Continuous Function. Expert Answer […]

### Find parametric equations for the path of a particle that moves along the circle

[x^2+(y-1)^2=4] In the manner describe: a) One around clockwise starting at $(2,1)$ b) Three times around counterclockwise starting at $(2,1)$ This question aims to understand the parametric equations and dependent and independent variables concepts. A sort of equation that uses an independent variable named a parameter (t) and in which dependent variables are described as […]

### Match the parametric equations with the graphs. Give reasons for your choices.

$(a) space x=t^4 -t+1 , y= t^2$ $(b) space x=t^2 -2t , y=sqrt t$ $(c) space x=sin2t ,y=sin ( t +sin 2t)$ $(d) space x=cos5t ,y=sin 2t$ $(e) space x=t+sin4t ,y= t^2 +cos3t$ $(f) space x=dfrac{sin2t }{4+t^2} ,y=dfrac{cos2t} {4+t^2}$ Graph I Graph II Graph III Graph IV Graph V Graph VI In this question, we […]

### Let C be the curve intersection of the parabolic cylinder x^2=2y and the surface 3z=xy. Find the exact length of C from the origin to the point (6,18,36).

This article aims to find the length of the curve $C$ from origin to point $(6,18,36)$. This article uses the concept of finding the length of arc length. The length of the curve defined by $f$ can be defined as the limit of the sum of lengths of linear segments for the regular partition $(a,b)$ as […]

### If f is continuous and integral 0 to 4 f(x)dx = 10 , find integral 0 to 2 f(2x)dx.

This problem aims to find the integral of a continuous function given an integral of the same function at some other point. This problem requires the knowledge of basic integration along with the integration substitution method. Expert Answer A continuous function is a function with no disruption in the variation of the function, and this […]

### Find the differential of each function. (a) y=tan (7t), (b) y=3-v^2/3+v^2

The main purpose of this question is to find the differential of each given function. A function is a fundamental mathematical concept that describes a relationship between a set of inputs and a set of possible outputs, with each input corresponding to one output. The input is an independent variable and the output is referred […]

### Solve the equation explicitly for y and differentiate to get y’ in terms of x.

(dfrac{1}{x}+dfrac{1}{y}=1). The main objective of this question is to explicitly write the given function in terms of $x$ and to express $y’$ by using explicit differentiation. An algebraic function in which the output variable, say a dependent variable, can be expressed explicitly in terms of the input variable, say an independent variable. This function typically […]