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# Basic Calculus

### Differential Equation Application

1) Find the equilibrium solution of the differential equation: Dy/dt = 3y(1-(1/2)y) Sketch the slope field and use it to determine whether each equilibrium is stable or unstable. 2) Consider the initial value problem Y^1 = 4 - y^2, y(0) = 1 Use the Euler's method with 5 steps to estimate y(1). Sketch the field and u

### Differential Equations and Normal Line

Question 1. 1) Find a vector normal to the surface z + 2xy = x2 + y2 at the point (1,1,0). 2) Determine if there are separable differential equations among the following ones and explain: a) dy/dx=sin(xy), b) dy/dx = (xy)/(X+y) c) dr/d(theta) = (r^2+1)cos(theta) 3) Find the general solution of the differential

### Multiple Variable Calculus

Question 1. Find the critical points of the function f(x, y) = 4xy2 - 4x2 - 2y2 and classify them as local maxima, minima or saddles or none of these. Question 2. The surface is defined by z = 3x2 + 2y2 - 3 Find the equation of the tangent plane to the surface at the point (1, 1, 2). Question 3. (

### Multiple Variable Calculus Questions

See attached document

### Vertical and Horizontal Asymptotes Determination

1.Find the asymptotes, the intercepts and sketch the following function. A computer sketch is not sufficient. You must explain.For each of the following functions, do a complete analysis: determine the vertical and horizontal asymptotes, and the x and y intercepts; and sketch a graph of the function. Section 6.2, number 6 y

### Comparison and Integral Tests

1. Apply the Basic Comparison Test (BCT) and Limit Comparison Test (LCT). Find an appropriate Comparison series to determine convergence or divergence. See attached file for formulas. 2. Apply the Integral Test: First use a method to see if the terms are decreasing, then to determine convergence or divergence. See attached f

### Volume of Solid from Revolution

Let R be the region bounded by the graphs of x - 2y - 8 = 0, and y=(x-6)/(x-3) a) Determine the VOLUME of the solid of revolution when R is revolved about the y-axis, using any suitable method. b) Also determine the volume of the solid revolution when R as defined above is revolved about the line y + 2 = 0, by any metho

### Quadratic Equations with Complex Roots

Determine the number and types of solutions for the following quadratic equations x2 - 5x = -6 Determine the value of c for which the following quadratic equations will have one root. (see attachment for the rest)

### Multiplying Brackets, Simplifying & Model Rocket Height

1. (a) Perform the indicated operation and express the result in the form a + bi. (1 - i) (3 + i). (b) Simplify. i15 (c) x - 2y - 7 = 3x i + 5yi - 10 i 2. A model rocket is launched with an initial velocity of 100 feet per second from the top of a hill which is 20 feet high. Determine the time, t , that the rocke

### Finding volume bounded by given curves

18. Compute the volume of the solid formed by revolving the region bounded by about (a) the x-axis; (b) y = 4. 20. Compute the volume of the solid formed by revolving the region bounded by and about (a) the y-axis; (b) x = 1. 4. Sketch the region, draw in a typical shell, identify the radius and height of the shell, an

### Solving a Set of Basic Mathematical Questions

See the attached file. 1. Write in radical notation and simplify if possible. 253/2 2. Assume all variables are positive real numbers. Write the following radicands with rational exponents and simplify if possible. 3. Solve for x (x + 2)(x - 2) = 15 + (x +1)(x - 7) Check your solution by substituting your solution

### Integral Calculus Volume Problems

I need help with these two problems. If you could please explain the solution, I would appreciate it. 1.Use cylindrical shells to compute the volume. The region bounded by y = x and y = x2 - 2, revolved about x = 3 2. A solid is formed by revolving the given region about the given line. Compute the volume exactly if possi

### Solving Transfer Function and Differential Equation in S Domain

Determine the transfer function (in the s domain) of a first order equation. (Laplace Transforms) Calculate the value of y(t) when t =10s.

### Using Reimann Sums

Use the 'midpoint' rule for calculating the integral of x^3 on the domain [6,7] using n=2,5, and 10 equally wide rectangles. Then calculate the integral on the domain [2,4].

### Calculus

Let f(x) = x^3, and compute the Riemann sum of f over the interval [6, 7], choosing the representative points to be the midpoints of the subintervals and using the following number of subintervals (n). (Round your answers to two decimal places.) (a) Two subintervals of equal length (n = 2) (b) Five subintervals of equal le

### Spherical Trigonometry Proof

For a rectangular sperical triangle, where C = 90 degrees, show that: tan a = sin b tan A and, tan b = sin a tan B This is how far I have a gotten: sin C = 1 cos C = 0 I am also aware that I should be using both the Law of Cosines and the Law of Sines. Using these, I was able to prove several other identities. I am co

### Solving Exact Differential Equations

(b) Find a suitable integrating factor R for the differential equation and verify whether the equation becomes exact after multiplying it by R. (c) Hence find the general solution of the differential equation. (See attached file for equations).

### Stiff Spring Motion

Question: The displacement s cm of the end of a stiff spring at time t seconds is given by: s = a (e^−kt) sin 2πft. Determine the velocity and acceleration of the end of the spring after 2 seconds if a = 3, k = 0.75 and f = 20.

### Minimize function

Suppose total transportation cost for a product can be approximated by the function: T(f) = 2.1f^2 - 25.9f +121.1 Where f is the number of facilities. Ignoring any other costs, find the optimal number of facilities with the minimum total transportation cost.

### LPP - determinin objective function and inequalities

Formulate but do not solve the following linear programming problem. A travel company decides to advertise in the Saturday travel sections of two major newspapers in town. The advertisements are directed at three groups of potential customers. Each advertisement in newspaper A is seen by 60,000 group I customers, 35,000 group

### Fourier series expansion - calculus

A function is defined as followed: see attachment Where f(t+2)=f(t) that is, f(t) has period 2. i) Draw a plot of the function f(t). Comment fully on whether the function is even or odd or none of these. ii) Find the first four non-zero coefficients for the Fourier series expansion of the function f(t) iii) Using eg. e

### Characteristics of increasing graph

Describe the following characteristics of the graph shown in Graph3.pdf: 1. Where is the function increasing, decreasing, or constant? 2. Are there any relative/absolute extrema? If so, where? 3. Is the graph smooth or choppy (piecewise)? 4. Are there any restrictions on the domain? 5. Are there any horizontal or

### Sketching the graph of a given function

Based on the information, create a sketch of the function on the axes provided. Please provide detailed explanation. ** Please see the attached file for the complete problem description ** -Increasing and constant: none -Decreasing: (-infinity, 2)... (please see the attached file) Thanks!

### Quarterly compounding of interest

Kurt Daniels wants to buy a \$30,000 car. He has saved \$27,000. Find the number of years it will take for his \$27,000 to grow to \$30,000 at 4% interest compound quarterly. Please provide a detailed explanation.

### Finding equations that represent exponential functions

Write an equation using the following information: Each represents an exponential function with base 2 or 3 translated and/or reflected 1) (-3,0), (-2,1), (0,7) Equation of horizontal asymptote: y=-1 2) (-1,3), (0,4), (-3,-3) Equation of horizontal asymptote: y=5

### Derivatives and continuity of functions

1. Find the derivative of the function. h(t) = t2(3t + 9)3 2. Find the derivative of the function. 3. Find the derivative of the function. Find the derivative of the function. 4. f(x) = ex + e−x ------------ 2 Find the derivative of

### Related Rates and calculus problems for real life situations

SEE ATTACHMENT FOR ALL PROBLEM QUESTIONS Related Rates Problems 1. A tiger escapes from a truck, right in front of the Empire State Building. I start running west along 34th Street at 2.5 m/s, while my friend takes off north on Fifth Avenue at 3 m/s. Draw a diagram of this situation. How fast is the distance between my

### Cooling of soup

A bowl of soup at 185 degree F is left in a room with a temperature of 75 degree F to cool. - 0.075^t : After t minutes, the temperature, T, of the soup is given T(t)= 75 + 110e Find the temperature of the soup 24 minutes after it is left in the room (round to the nearest degree).

### This Calculus Review on Integrals Problems contains problems of real dynamic situations like, German zeppelin bomber airship and spotlight, electrical circuit with parallel resistors having variable resistances, ideal gas law with changing pressure P, volume V and temperature T, fundamental theorem of calculus and Area enclosed by the two curves.

1. Evaluate the following inde finite integrals: 1. 1/x^2 dx 2. e^(-x) dx 3. sin(t) cos(t)dt 4. sqrt(s) ds 2. On a dark night in 1915, a German zeppelin bomber drifts menacingly over London. The men on the gro

### Pointwise limits f

Let fk (x) = sin (kx)/k k = 1, 2, 3, . . . . a) Determine the pointwise limit f of the sequence {fk} infinity k=1 on R. b) Show that the sequence {fk} infinity k=1 converges to f uniformly to f on R.