for |x| <1, the derivative of y=ln√1-x² is?

Question

for |x| <1, the derivative of y=ln√1-x² is
a. (x)/(1-x²)
b. (x)/ (x²-1)
c. (-1)/(x²-1)
d. (1)/2(1-x²)
e. (1)/ √1-x²

Answer 1

Let's solve the derivative and see what we get.

y = ln (sqrt(1 - x^2))

For one thing, you can use log properties to transform this into something simpler.

y = ln [ (1 - x^2)^(1/2) ]

Using the log property log[base b](a^c) = c*log[base b])(a),

y = (1/2) ln (1 - x^2)

Factoring the inside of the log,

y = (1/2) ln [(1 - x)(1 + x)]

Applying the log property log[base b](ac) = log[base b](a) + log[base b](c), we get:

y = (1/2) [ln (1 - x) + ln(1 + x)]

Keep in mind at that this point we haven't even taken the derivative yet; only simplified its form.

Now, taking the derivative, we get

y = (1/2) [ {1/(1 - x)}(-1) + 1/(1+x)}
y = (1/2) [1/(1 + x) - 1/(1 - x)]

We can change the form of this, to

y = (1/2) [ {1 - x} - {1 + x} ] / [(1 - x)(1 + x)]
y = (1/2) [-2x]/[(1 - x^2)]

Merging this into one term, we finally get

y = -x/(1 - x^2)

But we can factor out a -1 in the denominator.

y = -x / [-1(x^2 - 1)]

And now we can cancel the negative signs,

y = x / (x^2 - 1)

For that reason, the answer is (b).

Answer 2

b. It's easier to write the function as 1/2 ln(1-x^2).

Answer 3

b

Answer 4

answer is d

Answer 5

. Let f(x,y,z) = exyz + ln(1 + x2 + y2 + z2) where x,y,z are real numbers. What is the direction of maximum increase of f at the point (1,1,0)?


A.


B.


C.



D.


E.





solution


2. Let f(x,y) = (x2 + y2)-1/2 for (x,y) not equal to zero. What is the directional derivative of f at the point (x,y) in the direction toward the origin?


A. 1
B. (1/2)[(x+y)/(x2+y2)3/4]
C. 1/(x2+y2)3/2

D. (1/2)[(x+y)/(x2+y2)]
E. 1/(x2+y2)



solution


3. What is the directional derivative of f(x,y) = 4x2y4 - 2x + 5 at the point (2,1) in the direction <-3,4>?


A. -136/5
B. 107/[2(1073)1/2]
C. 160/7

D. 214/5
E. 214



solution


4. What is the directional derivative of f(x,y) = 5 - 4x2 - 3y at (x,y) toward (0,0)?


A. -8x - 3
B. (-8x2-3y)/(x2+y2)1/2
C. (-8x-3)/(64x2+9)1/2

D. 8x2 + 3y
E. (8x2+3y)/(x2+y2)1/2


solution

http://www.qu.edu.qa/home/myqu/falmuhannadi/Lecture%20Notes/Derivative%20Formulas.doc

In mathematics, a derivative is the rate of change of a quantity. A derivative is an instantaneous rate of change: calculated at a specific instant rather than as an average over time. The process of finding a derivative is called differentiation. The reverse process is integration. The two processes are the central concepts of calculus and are related via the fundamental theorem of calculus.

Moreover, for a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent to the graph of the function at that point. Derivatives can be used to characterize many properties of a function, including

whether and at what rate the function is increasing or decreasing through a value of the function
whether and where the function has maximum or minimum values.
The concept of a derivative can be extended to functions of more than one variable (see multivariable calculus), to functions of complex variables (see complex analysis) and to many other cases.

Differentiation has many applications throughout all numerate disciplines. For example, in physics, the derivative of the position of a moving body is its velocity and the second derivative of the body's position is its acceleration. In turn, the 'velocity' of the body in a given direction is its 'speed' in that direction, another derivative. Speed on position-time axes is the (instantaneous rate of) unit change in position of the body per unit change in time
Main article: History of calculus
The modern development of calculus is credited to Isaac Newton and Gottfried Leibniz who worked independently in the late 1600s.[1] Newton began his work in 1666 and Leibniz began his in 1676. However, Leibniz published his first paper in 1684, predating Newton's publication in 1693. There was a bitter controversy between the two men over who first invented calculus which shook the mathematical community in the early 18th century.


[edit] Differentiation and differentiability
Differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ (Delta) to refer to change in a quantity, this rate is defined as a limit of difference quotients


which means the limit as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written


suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The form "d y d x" is also used conversationally, although it may be confused with the notation for element of area.

Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. One precise way to define the derivative is as a limit [2]:


A function is differentiable at a point x if the above limit exists (as a finite real number) at that point. A function is differentiable on an interval if it is differentiable at every point within the interval.

As an alternative, the development of nonstandard analysis in the 20th century showed that Leibniz's original idea of the derivative as a ratio of infinitesimals can be made as rigorous as the formulation in terms of limits.

If a function is not continuous at a point, then there is no tangent line and the function is not differentiable at that point. However, even if a function is continuous at a point, it may not be differentiable there. For example, the function y = |x| is continuous at x = 0, but it is not differentiable there, due to the fact that the limit in the above definition does not exist (the limit from the right is 1 while the limit from the left is −1). Graphically, we see this as a "kink" in the graph at x = 0. Thus, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

The derivative of a function f at x is a quantity which varies if x varies. The derivative is therefore itself a function of x; there are several notations for this function, but f' is common.

The derivative of a derivative, if it exists, is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on. A function may have zero, a finite number, or an infinite number of derivatives.


[edit] Newton's difference quotient
The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line.


Tangent line at (x, f(x))
Secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)).To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is


This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:



Tangent line as limit of secants.If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x.

One cannot obtain the limit by substituting 0 for h, since it will result in division by zero. Instead, one must first modify the numerator to cancel h in the denominator. This process can be long and tedious for complicated functions, and many short cuts are commonly used which simplify the process.


[edit] Examples
Consider the graph of f(x) = 2x − 3. Using algebra and the Cartesian coordinate system, one can independently determine that this line has a slope of 2 at every point. And that is indeed the result one finds using the derivative. The slope at (4,5) is, using the above quotient:






The derivative and slope are equivalent.

Now consider the function f(x) = x2:







For any point x, the slope of the function f(x) = x2 is f'(x) = 2x.


[edit] Notations for differentiation

[edit] Lagrange's notation
The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

for the first derivative,
for the second derivative,
for the third derivative, and in general
for the nth derivative.


[edit] Leibniz's notation
The other common notation is Leibniz's notation for differentiation which is named after Gottfried Leibniz. For the function whose value at x is the derivative of f at x, we write:


With Leibniz's notation, we can write the derivative of f at the point a in two different ways:


If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as:


Higher derivatives are expressed as

or
for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:


which we can loosely write as:


Dropping brackets gives the notation above.

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember:


(In the formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In non-standard analysis, however, they can be viewed as infinitesimal numbers that cancel.)


[edit] Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name:



and so on.

Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives, and then, only to denote derivatives with respect to time, as opposed to other types of variables.


[edit] Euler's notation
Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator:

for the first derivative,
for the second derivative, and
for the nth derivative, provided n ≥ 2.

This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable:

for the first derivative,
for the second derivative, and
for the nth derivative, provided n ≥ 2.

Euler's notation is useful for stating and solving linear differential equations.


[edit] Critical points
Points on the graph of a function where the derivative is equal to zero or the derivative does not exist are called critical points or sometimes stationary points. If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points or endpoints. This is related to the extreme value theorem.


[edit] Physics
Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

Velocity is the derivative (with respect to time) of an object's displacement (distance from the original position).
Acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.
Jerk is the derivative (with respect to time) of an object's acceleration, that is, the third derivative (with respect to time) of an object's position, and second derivative (with respect to time) of an object's velocity.
For example, if an object's position on a curve is given by


then the object's velocity is


and the object's acceleration is


Since the acceleration is constant, the jerk of the object is zero.


[edit] Rules for finding the derivative
In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules.

Constant rule:
for any real number c
Constant multiple rule:
for any real number c (a consequence of the linearity rule below).
Linearity:
for all functions f and g and all real numbers a and b.
Power rule: If f(x) = xr, for some real number r;
.
Product rule:
for all functions f and g.
Quotient rule:
unless g is zero.
Chain rule: If f(x) = h(g(x)), then
.
Inverse function: If the function f(x) has an inverse g(x) = f − 1(x), then
.
In addition, the derivatives of some common functions are useful to know. See the table of derivatives.

As an example, the derivative of


is


The first term was calculated using the power rule, the second using the chain rule and the last two come from the product rule. The derivatives of sin(x), ln(x) and exp(x) can be found in table of derivatives.


[edit] Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x3 has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither.

In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3).

Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.


[edit] Generalizations
For more details on this topic, see derivative (generalizations).
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.

The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

In order to differentiate all continuous functions and much more, one defines the concept of distribution and weak derivatives.

For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions

http://www.intmath.com/Differentiation-transcendental/5_Derivative-logarithm.php

Sample Worksheet 1

MathExcel, Section 21
Name: _______________________
Worksheet 28
November 9, 1998

Groups should work cooperatively to complete as many of the following as possible. Any questions on problems should be asked by the group, not individuals.


Find all of the horizontal and vertical tangent lines to the following:
x2 + y2 = 2
3x2 + y2 = 12x

Show that the normal line to the circle x2+y2 = a2 at any point (s,t) passes through the origin. (The normal at any point on the curve is the line perpendicular to the tangent line.)

Show that the curves given by x2 + y2 = 4 and x - 2y = 0 are orthogonal. (Two curves are orthogonal when their tangent lines are perpindicular everywhere the curves interesect.)

Find an equation of the tangent line of the curve y2 = x3+3x2 at the point (1,-2). At what points does this curve have a horizontal tangent line? At what points does this curve have a vertical tangent line? After finding this points, use your calculator to graph the curve and check if your answers make sense. (Hint: write the curve as two functions and graph both functions at the same time.) Sketch the graph on your paper.

Find [(d2x)/( dy2)] if xy = cosy +2x

Find [dy/ dx]:
x = cos(xy)
y = tan23x

Find the slope of the tangent line and the normal line of x2+4xy+y2+3 = 0 at the point (2, -1).

Find the slope of the tangent line to the graph of the equation xy + sin(py) = x3 at the point (0,1).

Use implicit differentiation to determine the following derivatives. Answer in terms of x.

[d/ dx] arcsinx
[d/ dx] arctanx
[d/ dx] arcsec x
[d/ dx] arccsc x

Differentiate the following functions.

f(x) = arctan(2x)
h(t) = Ö{arcsint}
f(y) = earctany
j(x) = x2arccsc(Öx)

The derivative of an inverse of a function can be found by using the formula
(f-1)¢(x) = [1/( f¢(f-1(x)))]
Prove this formula in two ways:

By a property of inverses f(f-1)(x) = x. Differentiate both sides of this equation and then solve for (f-1)¢(x).
Let y = f-1(x). Then, by the definition of an inverse function, f(y) = x. Differentiate this function implicitly, and when you have solved for [dy/ dx], you will have proven the formula.

If f(x) = ex, then f-1(x) = ln(x). Use this an the formula above to find the derivative of ln(x).

Verify the formula [d/ dx](ax) = ax ln(a) using that [d/ dx](ex) = ex, chain rule and properties of logs.

Use implicit differentiation, the inverse formula or the log change of base formula to find the derivative of y = logb(x).

Differentiate the following functions:
f(x) = ln(1+x2)
g(t) = log3(x2+1)
h(x) = sin(4x)ln(3x)
g(t) = e3x ln(15)

Differentiate the following. (Hint: use properties of logs to simplify the problem BEFORE taking the derivative.)
g(x) = ln(x4 (x3-4)2 (4x-5))
h(x) = 3ln( [(3-x7)/( (x-2)3 (64+3x)9)]
k(x) = ln(sin(x) cos(x) (3x-4)4)

Derivatives of Power Functions and Polynomials.

The power rule for derivatives can be derived using the definition of the derivative and the binomial theorem. The result is the following theorem:

If f(x) = xn then f '(x) = nxn-1

Polynomials are sums of power functions. In order to obtain their derivatives, we need to establish the following facts:



where c is independent of x, and



While these rules are being applied to power functions and polynomials first, they work for any functions.

Examples:

1. A polynomial with positive and negative exponents.



In this example, we rewrote the rational terms as power terms with negative exponents.

2. A rational function with a single term in the denominator.



In this example, we split the fraction and simplified before taking the derivative. This is often simpler than employing the quotient rule.

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3. A function with a radical term.



In this example, we rewrote the radical in terms of a rational exponent. The last result is what we obtain when we find the derivative using the definition of the derivative.

4. Another function with more complex radical terms.



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Derivatives of Exponential Functions.

From the definition of the derivative, we can deduce that



Examples:

1. The derivative of 2 x.



2. The derivative of 5(4.6)x.



3. The derivative of (ln3)x.



4. The derivative of e x.



This last result is the consequence of the fact that ln e = 1.

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The Product Rule

When a function is the product of two functions, or can be deconvolved as such a product, then the following theorem allows us to find its derivative:

If y = f(x)g(x), then [f(x)g(x)]’ = f’(x) g(x) + f(x) g’(x)

Another way to write this, using Leibniz' notation is



Notice that the derivative of a product of functions is not just the product of their derivatives; the derivative is somewhat more complex. That is,



Instead, we have



Examples:

1. The product f(x) = (2x + 1)(x2 - 2).



You should verify that if you had FOILed the product first, then taken the derivative, the result would have been the same.

2. Another example:



Factoring the result at the end is usually convenient for algebraically finding critical points and intervals of increase/decrease.

3. An example of numerically defined functions. Let f(2) = 1, f '(2) = -1, g(2) = -3, and g'(2) = 4. Find the derivative of f(x)g(x) at x = 2.

Solution: By the product rule, the derivative of the product of f and g at x = 2 is

f '(2)g(2) + f(2)g'(2) = (-1)(-3) + (1)(4) = 7

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The Quotient Rule:

When a function is the quotient of two functions, or can be deconvolved as such a quotient, then the following theorem allows us to find its derivative:

If y = f(x)/g(x),



Another way to write this, using Leibniz' notation is



Notice that the derivative of a quotient of functions is not just the quotient of their derivatives; the derivative is somewhat more complex. That is,



Instead, we have



Examples:

1. A rational function of polynomials:



You should verify that this is the result we would obtain if we had algebraically simplified the rational function first, then taken the derivative.

2. Another rational function which cannot be algebraically simplified:



3. An example of numerically defined functions. Let f(2) = 1, f '(2) = -1, g(2) = -3, and g'(2) = 4. Find the derivative of f(x)/g(x) at x = 2.

Solution: By the quotient rule, the derivative of the product of f and g at x = 2 is



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The Chain Rule

If y = f(u) and u = g(x), and the derivatives of f and g exist, then the composed function defined by y = f(g(x)) has a derivative given by



This is the chain rule. It requires these steps:

· For a composite function, write f as a function of u, and u itself a function of x.

· Find the derivatives of f and u.

· Apply the chain rule to find the derivative of the composite function.

It is convenient and easy to think of composed functions as an "outside" function and and "inside" function. We take the derivative of the outside function, leaving the inside function alone. This is the f '(g(x)) part. Then we multiply the result by the derivative of the inside function. This is the g'(x) part.

Notice that the derivative of composed functions is not merely the composed derivatives:



Rather,



Here, ex is the "outside function" and x2 is the "inside" function.

Examples:

1. A polynomial to a power.



2. If f(x) = e x and g(x) = x 2 + 3x, find f(g(x)) and f (g(x))¢.

Solution:



3. An example of numerically defined functions: If f(2) = 5, f¢ (2) = -1, g(1) = 2, and g¢ (1) = 4, find f (g(1)) and f (g(1))¢.

Solution using the meaning of composed functions, the chain rule and the given data:



4. The hot glowing gas of a star radiates energy according to the function E = sT 4.If T is a function of the radius of the star, use the Chain Rule to find a formula for dE/dr.

Solution, using the chain rule:



In this case, we do not have an expression for dT/dr, so we leave the derivative of T alone. We encounter this type of situation often when solving related rates problems. This reveals that E is really an implicit function of r, that is, it depends on r, but the manner in which it depends on r may not be obvious, or explicit.

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Trigonometric Functions

The definition of the derivative and the addition formulas for sine and cosine can be used to derive the following theorems:

If f(x) = sin x then f¢ (x) = cos x

If f(x) = cos x then f¢(x) = -sin x

Using the tangent identity and the quotient rule, the following theorem is derived:



Similarly, the following derivatives can be derived:



Examples.

1. Find the derivative of y = 2sin x + 3tan x.

Solution: Using the above theorems, y' = 2cos x + 3sec2 x

2. A population of foxes varies seasonally according to the model



where P is in thousands and t is in months since Jan 1. Find a function that models the rate of change of the fox population with respect to time.

Solution: Using the above theorem for the derivative of cosine and the chain rule:



3. Show that the functional side of the Pythagorean Identity is indeed constant for all x.

Solution: the Pythagorean Identity is sin2x + cos2x = 1. Taking the derivative of the left side of the equation gives us:

2sinxcosx + 2cosx(-sinx) = 2sinxcosx - 2sinxcosx = 0

Since the derivative equals zero, the rate of change is zero (for all x), hence the identity is a constant.

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Inverse Functions

The derivatives of various other functions can be obtained by the chain rule and the composition of inverse functions. The method goes like this:

Suppose f(x) is differentiable and its inverse function f -1(x) is also differentiable. When they are composed, the result is

f -1(f(x)) = x.

To find the derivative of f(x), the derivative of the composed functions, obtained via the chain rule, yields the following result:



Suppose f(x) = ln x. Then f -1(x) = e x. Then f -1(f (x))’ = e ln x = x. Thus, the following:



Other derivatives obtainable by this method include:



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Hyperbolic Functions

The hyperbolic functions are defined as



Their derivatives are obtainable by the rules outlined above:



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Parametric Functions

Parametric equations describe the motion of a point P as independent functions of the parameter t as it wanders about the xy-plane. The x- and y-coordinates are

x f(t) and y = g(t)

Parametric curves come in mind-boggling variety. Any choice of two such equations and a t-interval produces a parametric curve. Often the result is beautiful, useful, interesting, or all three.

The rate of change of position, P = (f(t), g(t)) along a parametrically defined curve is given by



This is often thought of as the velocity of the particle since for many applications the independent variable t represents time. The slope of the parametric curve is given by



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Polar Equations

The location of a point on the xy-plane can be expressed in polar form, that is, in terms of its distance (radius, r) from the origin and the angle, q, its radius vector makes with the positive x-axis. Cartesian (rectangular) graphs are often represented y = f(x). Such functions can also be represented in polar form r = f(q). Conversion of points or equations between systems are via the transformation equations:



The derivatives of x and y with respect to q are obtainable by the product rule:



Thus the slope of a polar curve is given by



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Vectors

Vectors are quantities that have both magnitude and direction. For a vector of the form



its magnitude is



and its direction with respect to the positive x-axis is



As such, vectors are really parametric functions. Therefore, the rate of change of a vector is determined determining the rates of change for its components:

Let f(x) = 1/(1+x) for x -1. What is the nth derivative, f(n)(x)?


A. n!(1+x)n+1
B. -n!/(1+x)n+1
C. n!/(1+x)n+1

D. -n!/(1+x)n
E. (-1)nn!/(1+x)n+1



solution


2. If f(x) = (x e3x)/(1 + x) for x 1, then f(1) = ?


A. 3e3/4
B. 5e3/4
C. 7e3/4

D. 9e3/4
E. 4e3



solution


3. If u = (x2+9)1/2 and v = 3x2-2x, then what is du/dv as a function of x?


A. 1/(4(3x-1)(x2+9)1/2) for x 1/3
B. (3x-1)/(2(x2+9)1/2)
C. (2x(3x-1))/(x2+9)1/2

D. x/(2(3x-1)(x2+9)1/2) for x 1/3




solution




4. The volume (in gallons) of water in a tank after t hours is given by f(t) = 600 sin2(Pi*t/12) for 0 ú t ú 6. What is the rate of flow of water into the tank, in gallons per hour?


A.100(Pi)sin(Pi*t/12)
B.100(Pi)cos(Pi*t/12)sin(Pi*t/12)

C.1200cos(Pi*t/12)sin(Pi*t/12)
D.50(Pi)cos2(Pi*t/12)

E.600cos2(Pi*t/12)


solution


5. What is the slope of the line tangent to the curve y3 - x2y + 6 = 0 at the point (1,-2)?


A. -(2/5)
B. -(4/11)
C. 4/11

D. 11/4
E. 8



solution



6. If f(x) = xex/sin(x) for 0 < x < p , then f(x) =

A. ex/cosx
B. ex(x+1)/cosx

C.ex[sinx+xcosx/sin2x]
D.ex[x(sinx+cosx)+sinx]/sin2x

E.ex[x(sinx-cosx)+sinx]/sin2x



solution




7. What is the slope of the tangent line to the curve x3 + y3 - 3xy = 13 at the point (2,-1)?


A. -5
B. -4
C. -1/5

D. 4
E. 5



solution


8. Let f(x) = x2 ln x - (x3 + 3)/2x for x > 0. Then f(x) =


A. 2 - (3x2/2)
B. 2x ln x + x - (3x2/2)
C. 2x ln x + (3/2x2)

D. 2x ln x + 2x - (3/2x2)
E. 2 - x + (3/2x2)



solution


9. Let f(x) = 2sin x. Then f"(0) =


A. -ln 2
B. -(ln 2)2
C. 0

D. (ln 2)2
E. ln 2



solution




10. Let f(x) = sin(2x + 1) and g(x) = x3 + 3 for all real x. Which of the following is equal to the derivative of the composite function f[g(x)]?


A. sin(2x3 + 7)
B. cos(2x3 + 7)
C. 6 sin(2x3 + 7)

D. 6 cos(2x3 + 7)
E. 6x2 cos(2x3 + 7)



solution


11. The equation y2 exy = 9 e-3 x2 defines y as a differentiable function of x. What is the value of dy/dx for x = -1, y =3?


A. -15/2
B. -9/2
C. -3/2

D. 6
E. 15



solution


12. A point moves along a number line so that its position at time t >= 0 is s(t) = 2t3 - 15t2 + 36t - 10. What is the position of the point when it first changes direction?


A. -63
B. 0
C. 17

D. 18
E. 98



solution


13. What is the derivative of f(x) = sin(xx) for x > 0?


A. xx(1 + ln x) cos(xx)
B. xx cos(xx)
C. xx ln x cos(xx)

D. cos(xx)
E. cos(xx ln x)



solution


14. What is the derivative of f(x) = 4x(x2 + 1)3 ?


A.4(x2 + 1)2(x2 + 3x + 1)
B. 4(x2 + 1)2(7x2 + 1)
C. 8x2(x2 + 1)3

D. 12x(x2 + 1)2
E. 24x2(x2 + 1)2



solution


15. Which of the following functions is continuous everywhere, but has at least one point where it is not differentiable?


A. tan x
B. |x|/x
C. sin x

D. e-x
E. x1/3



solution


16. What is the slope of the line tangent to the circle x2 + y2 = 45 at the point (6,-3)?


A. -6
B. -1/2
C. 1/6

D. 2
E. 11/2

http://www.math.dartmouth.edu/~calcsite/video1.html

www.maths.tcd.ie/~odunlain/1ba1/tests/99002ans.pdf

home.earthlink.net/~djbach/calc.html

http://www.mathgoodies.com/forums/topic.asp?
TOPIC_ID=31402&

personal.ecu.edu/shlapentokha/courses/2122/Fall03/final.pdf -

www.math.mcmaster.ca/riosc/M1K03/FinalK01sampssolutions.pdf

http://www.ltcconline.net/greenl/Courses/106/expLogTrig/logs.htm

www-math.cudenver.edu/~rbyrne/online/140w8f.htm

ems.calumet.purdue.edu/mcss/.../ma223/chapter2/derivativepracone.html

http://users.wpi.edu/~vadim/NM_I/Projects/add-problems.html