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# Derivatives

### Method of the characteristics

Hi. I have some questions of (PDEs). I have attached the document in attachments. But, please the solution must has the solutions' steps. Note: the first question has simple change(See that in attachements )

### Account Balance using Partial Derivative

Let B(r,t) compute the balance (in dollars) of account after t years with an r% interest rate. a) What are the units of dB/dr? (dB/dr is in partials notation) b) Give a real world interpretation of dB/dr evaluated at (5,10) = 56. c) If B(5,10) = 5000 and dB/dt evaluated at (5,10) = 165, estimate B(4,11). (dB/dt is parti

### Directional derivative

Let f(x) = e^(x+y+3z a.) Find the directional derivative at P(-2,2,-1) in the direction of Q(18,-2,4) b.) Find the direction of maximum decrease at P. (your answer does not have to be unit vector.) What is the maximum rate of decrease?

### Basic Differentiation Formulas

Differentiate the function: (2) f(x)=sq root (30) (4) F(x) = -4x^10 (6) f(t)=(0.5t^6) - (3t^4) + t (14) R(x)= (sq root (10))/(x^7) (18) g(u)= (sq root (2u)) + (sq root (3u)) (22) y = sin theta/2 + c/theta

### Basic Differenitation Formulas

Use the definition of the derivative to find the derivatives of the following functions: (a) f(x) = 4 (b) Think through the steps needed to do this problem. Would the answer be the same for f(x)=5? f(x)=c for any constant c? Why or why not? (c) Think through the steps needed to do this problem. Can you write down the steps

### Derivatives and Rates of Change

(12) If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters) after t seconds is given by H = 58t - 0.83t^2. (a) Find the velocity of the arrow after one second. (b) Find the velocity of the arrow when t = a. (c) When will the arrow hit the moon? (d) With what velocity will the arrow hit the

### Differentiation

1. Find the rate of change dy/dx where x = x0 (Compute the derivative of the function from the definition only, using limits. Show all steps.) y = 1/(2-x), x0 = -3 2. Differentiate the function. Simplify your answer. f(x) = (1/4)x^8 - (1/2)x^6 - x +2 3. Find dy/dx by implicit differentiation. y^2 +3xy -4x^2 =

### product rule simplification

Please provide step by step solution. Thank you! Algebra Chapter 09 1. Use the product rule to simplify âˆš12x^8 2. Evaluate the expression 25-3/2 3. Simplify the expression âˆš2t5 * âˆš10t4 4. Divide and simplify âˆš14/âˆš7 5. Solve and check for extraneous solutions âˆša-1-5=1 Chapter

### Derivatives, Tangents and Rate of Change

All solutions must be detailed and the final answers simplified. Show all work! 1. Differentiate the given function. Simplify your answer. 2. Differentiate the given function. Simplify your answer. 3. Differentiate the given function. Simplify your answer. 4. Find the equation of the line that is ta

### Rate of Change..

If \$10,000 is invested at an annual rate r(expressed as decimal) compound weekly, the total amount (principal P and interest) accumulated after 10 years is given by the formula. A=10,000(1+r/52)^520 A.Find the rate of change of A with respect to r. B.Find the percentage rate of change of A with respe

### Tangent Lines: Example Problem

Find an equation for the tangent line at the given curve at the point where x=x0; a. y=(x^2+3x-1)(2-x); x0=1 b. y=x+7/5-2x; x0;=0 Find all points on the given graph of the given function where the tangent line is horizontal. f(x)=(x-1)(x^2-8x+7)

### Derivatives Exponential and Natural Lograithmic Functions

Differentiate the functions in these problems. 1. f(x) = e^(sqrt(x)) + e^(-(sqrt)(x)) 2. f(x) = (sqrt)(e^2x + e^-2x) 3. f(x) = ln (4-x^2) 4. f(x) = ln [(1+x)^2] 5. f(x) = ln (sin^2x) 6. f(x) = (lnx)^3

### Function, derivatives, error, and stationary points.

See attached. Consider the function of two real variables x and y (y not 1) defined by.... Find the first-order and second-order partial derivatives of f. Determine the second-order Taylor polynomial for f near (0,-1) Use first-order partial derivatives to determine the least and greatest possible values. Use the chain ru

### Differentiate the functions

Please see the attached file for the fully formatted problems. Differentiate the functions. 1. g(t) = ( )1/2 g(t) = (3t5)-1/2 g(t) = 3t-5/2 g'(t) = t-7/2 2. ****There is a closing bracket after 4. Where is the open bracket? It is necessary to see where the open bracket is located is its location will provo

### Concavity, derivations, and proofs

Please see attachment All the questions Determine whether....converge or diverge ...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove... Using binomial coefficients, derive a formula for the nth derivative of the product of two functions. Suppose that f(x) has

### Use the rule of 78s to find the amount of interest saved.

A \$400 loan is to be paid off in 66 monthly payments of \$11.62. The borrower decides to pay off the loan after 18 payments. Use the rule of 78s to find the amount of interest saved.

### Logarithmic Differentiation to find the Derivative

Use logarithmic differentiation to find the derivative f(x)= e to the 2x power (2x- 1) to the 5th power / (x to the 3rd power + 5 to the second power times (4-7x) Applied Calculus 9th edition by Laurence D Hoffman and Gerald L. Bradley

### Profit Function and Maximum Profit

A manufacturer finds that the total profit from producing and selling Q units of a product is given by the profit function: Total Profit = f(Q) = - 460 + 100Q - Q^2 1. Compute the value of the function at Q=10 Total Profit = f(10)= - 460 + 100(10) - 10^2 Total Profit = f(10)= - 460

### Vectors, Planes and Partial Derivatives

1. Let a = 2i + 3j and b = -9i + 6j. Find c = a - b. A) c = -3j B) c = 9i C) c = 11i + 9j D) c = 11i - 3j 2. Let a = 2i + 3j and b = -9i + 6j. Find d = a ? b. A) 36 B) -36 C) 0 D) -18i2 + 18j2 3. Find the intersection of L1: x - 2 = Â½(y + 1) = 1/3(z - 3), L2: 1/3(x - 5) = Â½(y - 1) = z - 4, if they

### First and Second Derivatives and Minimizing

A company holds spare parts for its car maintenance service. There is a steady demand for these parts. If the company orders large numbers once a year, then they have to pay considerable warehouse costs to stock them. If they order small numbers very frequently then they have to pay considerable admin costs for processing all th

### Implicit Differentiation, Slope of a Curve and Equation

Use implicit differentiation to find the slope of the curve : x^3 - 3xy + y^4 = 5x at (0.5, -0.977). Find the equation of the tangent line at this point. Please see the attached file for the fully formatted problems.

### Partial Derivitives and the Chain Rule

Let F(u,v) be a function of two variables. Find f '(x) for each of the following. Use F_u and F_v for F_u and F_v (a) f(x) = F(x, 6). f '(x) = (b) f(x) = F(3, x). f '(x) = (c) f(x) = F(x, x). f '(x) =

### Partial derivatives and chain rule

Find (partial z)/(partial u)( and ) (partial z)/(partial v) using the chain rule. Assume the variables are restricted to domains on which the functions are defined. Your answers should be in terms of u and v. z=arctan(x/y) x=u^2+v^2 y=u^2-v^2 partial z/partial u = Partial z/partial v =

### Derivatives, Limits and Equation of Tangent Lines

1. Use the limit definition to find f'(x) for f(x) = ? (show all work). 2. Write the limit definition of derivative: f'(x) = lim 3. For the function f(x) = , use the limit definition of derivative to find the slope of the tangent line to the function at x=4. Show all steps for full credit. (b) Find the equatio

### Quotient Power and Exponent Rules

Please help explain the following questions and provide an example: 1. explain the quotient rule for exponents and give an example. 2. explain the power rule for exponents and give an example. 3. explain the negative exponent rule and give an example. 4. explain the zero exponent rule and give an example.

### Inflation Rate and "Rule of 70"

If the CPI was 110 last year and is 121 this year, what is this year's rate of inflation? What is the "rule of 70"? How long would it take for the price level to double if inflation persisted at (a) 2, (b) 5, and (c) 10 percent per year?

### Calculus Problems : Continuity, Limits, Derivatives, Inequalities, Quadratic Equations and Parabolas and Maximum Height

1. Solve log&#8326;x-3=0 2. A business owner is comparing the costs of purchasing inventory and the profit from the sale of the product. The relationship proves to be linear. Which type of variation will describe the data? Direct as nth power joint regress Inverse direct 3. Solve xÂ²-25<0 4. What is the

### Applications of Derivatives

An isosceles triangle whose base is the interval from (0,0) to (c,0) has its vertex on the graph of f, where f(x)=12-x^2 for x is greater than or equal to 0 and f(x) is greater than or equal to 0. For what value of c does the triangle have maximum area? Justify your answer.

### Finding Critical Points of a Function

See attachment. Find all critical points of the function: f (x,y) = x^3 + y^3 - 4x - 9y + 17 Classify the critical point as a relative minimum, relative maximum, or saddle point using the second derivative test.

### Differential Equation : Solution

Verify that the function is a solution of the differential equation: y = 10e^(-3t) y' + 3y = 0