Author Archives: Engineering Math

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 9

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PROBLEM:

If $\displaystyle f\left(x\right)=3x^2-4x+1$ , find $\displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0$ .

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SOLUTION:

\begin{align*} \frac{f\left(h+3\right)-f\left(3\right)}{h} & =\frac{\left[3\left(h+3\right)^2-4\left(h+3\right)+1\right]-\left[3\left(3\right)^2-4\left(3\right)+1\right]}{h} \\ & =\frac{3\left(h^2+6h+9\right)-4h-12+1-16}{h}\\ & =\frac{3h^2+18h+27-4h-12+1-16}{h}\\ & =\frac{3h^2+14h}{h}\\ & =\frac{h\left(3h+14\right)}{h}\\ & =3h+14 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 8

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PROBLEM:

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

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SOLUTION:

\begin{align*} \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h} & =\frac{\left[\left(x+h\right)^2+1\right]-\left(x^2+1\right)\:}{h}\\ \\ & =\frac{x^2+2xh+h^2+1-x^2-1}{h}\\ \\ & =\frac{2xh+h^2}{h}\\ \\ & =\frac{h\left(2x+h\right)}{h}\\ \\ & =2x+h \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 7

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PROBLEM:

A right circular cylinder, a radius of base $x$ , height $y$ , is inscribed in a right circular cone, radius of base $r$ and a height $h$ . Express $y$ as a function of $x$ ( $r$ and $h$ are constants).

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SOLUTION:

Refer to the figure below for an elevation view.

Schematic Diagram of a right circular cylinder inscribed in a right circular cone. — Diagram of a right circular cylinder with a base radius of r and height y inscribed in a right circular cone with base radius r and height h.

By ratio and proportion of two similar triangles, we have

\begin{align*} \frac{y}{r-x} & = \frac{h}{r} \\ y & =\frac{h\left(r-x\right)\:}{r} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 6

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PROBLEM:

The stiffness of a beam of rectangular cross-section is proportional to the breadth and the cube of the depth. If the breadth is 20 cm, express the stiffness as a function of the depth.

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SOLUTION:

Let $S$ =stiffness, $b$ =breadth, and $d$ =depth

\begin{align*} S & =bd^3 \\ S & = 20 d^3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 5

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PROBLEM:

Express the area $A$ of an equilateral triangle as a function of its side $x$ .

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SOLUTION:

From the formula of the area of a triangle, $\displaystyle A=\frac{1}{2} \text{a}\text{b} \sin\left(\theta \right)$ . Also, we know that the interior angle of an equilateral triangle is 60 degrees, and $\displaystyle \sin\:60^{\circ} =\frac{\sqrt{3}}{2}$ .

\begin{align*} A & =\frac{1}{2} \text{a}\text{b} \sin\left(\theta \right) \\ A & =\frac{1}{2} \cdot x\cdot x\cdot \sin\:60^{\circ} \\ A & =\frac{1}{2}\cdot x^2\cdot \frac{\sqrt{3}}{2} \\ A & =\frac{\sqrt{3}}{4}x^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 4

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PROBLEM:

Express the distance $D$ traveled in $t$ hr by a car whose speed is 60 km/hr.

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SOLUTION:

\begin{align*} \text{Distance} & = \text{Rate} \times \text{Time} \\ D & =\left(60\:\text{km/hr} \right)\cdot t \ \text{hr} \\ D & =60t \ \text{km} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 3

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PROBLEM:

If $\displaystyle y= \tan\left(x+\pi \right)$ , find $x$ as a function of $y$ .

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SOLUTION:

\begin{align*} y & = \tan\left(x+\pi \right) \\ x+\pi & = \tan^{-1}y \\ x & = \tan^{-1}y-\pi \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 2

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PROBLEM:

If $\displaystyle y=\frac{x^2+3}{x}$ , find $x$ as a function of $y$ .

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SOLUTION:

\begin{align*} y & = \frac{x^2+3}{x} \\ xy & =x^2+3 \\ x^2-xy+3&=0 \end{align*}

Solve for $x$ using the quadratic formula. We have $a=1,\:b=-y,\:\text{and}\:c=3$

\begin{align*} x & =\frac{-b\pm \sqrt{b^2-4ac}\:}{2a} \\ x & =\frac{ -\left(-y\right)\pm \sqrt{\left(-y\right)^2-4\left(1\right)\left(3\right)}}{2\left(1\right)} \\ x & =\frac{y\pm \sqrt{y^2-12}}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 1

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PROBLEM:

If $\displaystyle f\left(x\right)=x^2-4x$ , find

a) $\displaystyle f\left(-5\right)$

b) $\displaystyle f\left(y^2+1\right)$

c) $\displaystyle f\left(x+\Delta x\right)$

d) $\displaystyle f\left(x+1\right)-f\left(x-1\right)$

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SOLUTION:

Part A

\begin{align*} f\left(-5\right) & =\left(-5\right)^2-4\left(-5\right)\\ & =25+20\\ & =45 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

\begin{align*} f\left(y^2+1\right) & = \left(y^2+1\right)^2-4\left(y^2+1\right)\\ & =y^4+2y^2+1-4y^2-4\\ & =y^4-2y^2-3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part C

\begin{align*} f\left(x+\Delta x\right)&=\left(x+\Delta x\right)^2-4\left(x+\Delta x\right)\\ & =\left(x+\Delta x\right)\left[\left(x+\Delta x\right)-4\right]\\ & =\left(x+\Delta x\right)\left(x+\Delta x-4\right) \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

Part D

\begin{align*} f\left(x+1\right)-f\left(x-1\right) & =\left[\left(x+1\right)^2-4\left(x+1\right)\right]-\left[\left(x-1\right)^2-4\left(x-1\right)\right]\\ & = \left[x^2+2x+1-4x-4\right]-\left[x^2-2x+1-4x+4\right]\\ & =x^2-x^2+2x-4x+2x+4x+1-4-1-4\\ & =4x-8\\ & =4\left(x-2\right) \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 2 Problem 31

A swan on a lake gets airborne by flapping its wings and running on top of the water.

(a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.350 m/s², how far will it travel before becoming airborne?

(b) How long does this take?

Solution:

Part A

We are given the following values: $v_f=6.00\:\text{m/s}$ ; $v_0=0\:\text{m/s}$ ; and $a=0.350\:\text{m/s}^2$ .

From the kinematic equations, the most applicable formula to solve for the change in distance, $\Delta \text{x}$ , is

\left(v_f\right)^2=\left(v_0\right)^2+2a\Delta x

Solving for $\Delta \text{x}$ in terms of the other variables, we have

\Delta x=\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2a}

Substituting the given values,

\begin{align*} \Delta x & =\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2a} \\ \Delta x & =\frac{\left(6.00\:\text{m/s}\right)^2-\left(0.00\:\text{m/s}\right)^2}{2\left(0.350\:\text{m/s}^2\right)} \\ \Delta x & =51.4286\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

From the formula $v_f=v_0+at$ , solve for time, $t$ in terms of the other variables.

t=\frac{v_f-v_0}{a}

Substitute the given values

\begin{align*} t & =\frac{v_f-v_0}{a} \\ t & =\frac{6.00\:\text{m/s}-0.00\:\text{m/s}}{0.350\:\text{m/s}^2} \\ t & =17.1429\:\text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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Author Archives: Engineering Math

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 9

PROBLEM:

If $\displaystyle f\left(x\right)=3x^2-4x+1$ , find $\displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 8

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 7

A right circular cylinder, a radius of base $x$ , height $y$ , is inscribed in a right circular cone, radius of base $r$ and a height $h$ . Express $y$ as a function of $x$ ( $r$ and $h$ are constants).

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 6

The stiffness of a beam of rectangular cross-section is proportional to the breadth and the cube of the depth. If the breadth is 20 cm, express the stiffness as a function of the depth.

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 5

Express the area $A$ of an equilateral triangle as a function of its side $x$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 4

Express the distance $D$ traveled in $t$ hr by a car whose speed is 60 km/hr.

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 3

If $\displaystyle y= \tan\left(x+\pi \right)$ , find $x$ as a function of $y$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 2

If $\displaystyle y=\frac{x^2+3}{x}$ , find $x$ as a function of $y$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 1

If $\displaystyle f\left(x\right)=x^2-4x$ , find

a) $\displaystyle f\left(-5\right)$

b) $\displaystyle f\left(y^2+1\right)$

c) $\displaystyle f\left(x+\Delta x\right)$

d) $\displaystyle f\left(x+1\right)-f\left(x-1\right)$

Part A

Part B

Part C

Part D

College Physics by Openstax Chapter 2 Problem 31

A swan on a lake gets airborne by flapping its wings and running on top of the water.

(a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.350 m/s², how far will it travel before becoming airborne?

(b) How long does this take?

Part B

PROBLEM:

If \displaystyle f\left(x\right)=3x^2-4x+1, find \displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0.

If \displaystyle f\left(x\right)=x^2+1, find \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0.

A right circular cylinder, a radius of base x, height y, is inscribed in a right circular cone, radius of base r and a height h. Express y as a function of x (r and h are constants).

The stiffness of a beam of rectangular cross-section is proportional to the breadth and the cube of the depth. If the breadth is 20 cm, express the stiffness as a function of the depth.

Express the area A of an equilateral triangle as a function of its side x.

Express the distance D traveled in t hr by a car whose speed is 60 km/hr.

If \displaystyle y= \tan\left(x+\pi \right), find x as a function of y.

If \displaystyle y=\frac{x^2+3}{x}, find x as a function of y.

If \displaystyle f\left(x\right)=x^2-4x, find

a) \displaystyle f\left(-5\right)

b) \displaystyle f\left(y^2+1\right)

c) \displaystyle f\left(x+\Delta x\right)

d) \displaystyle f\left(x+1\right)-f\left(x-1\right)

Part A

Part B

Part C

Part D

A swan on a lake gets airborne by flapping its wings and running on top of the water.

(a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.350 m/s2, how far will it travel before becoming airborne?

(b) How long does this take?

Part B

If $\displaystyle f\left(x\right)=3x^2-4x+1$ , find $\displaystyle \frac{f\left(h+3\right)-f\left(3\right)}{h},\:h\ne 0$ .

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

A right circular cylinder, a radius of base $x$ , height $y$ , is inscribed in a right circular cone, radius of base $r$ and a height $h$ . Express $y$ as a function of $x$ ( $r$ and $h$ are constants).

Express the area $A$ of an equilateral triangle as a function of its side $x$ .

Express the distance $D$ traveled in $t$ hr by a car whose speed is 60 km/hr.

If $\displaystyle y= \tan\left(x+\pi \right)$ , find $x$ as a function of $y$ .

If $\displaystyle y=\frac{x^2+3}{x}$ , find $x$ as a function of $y$ .

If $\displaystyle f\left(x\right)=x^2-4x$ , find

a) $\displaystyle f\left(-5\right)$

b) $\displaystyle f\left(y^2+1\right)$

c) $\displaystyle f\left(x+\Delta x\right)$

d) $\displaystyle f\left(x+1\right)-f\left(x-1\right)$

(a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.350 m/s², how far will it travel before becoming airborne?