MATH241 Lecture Notes - Lecture 17: Implicit Function

32 views5 pages

cerisegiraffe538

9 May 2018

School

University of Delaware

Department

Mathematics

Course

MATH241

Professor

Mark Adrian

For unlimited access to Class Notes, a Class+ subscription is required.

MATH241 - Lecture 17 - Implicit Differentiation

3.5: Implicit Differentiation

Explicit Form

Express one variable in terms of the other variable

(x)y=f

y= 3x2+e5x

an2xy =√x3+ 4 + t

Implicit Form

The variables are defined in terms of an equation such as:

6x2+y2= 3

lm xyex+ey= 2

In some cases it is possible to obtain an explicit expression for one variable in terms of the

other variable

Example



Find an explicit expression for in terms of y x

6x2+y2= 3

Solution:

±y=√36 − x2

Most of the time we cannot obtain an explicit expression from an implicit expression, as in

lm xyex+ey= 2

Implicit Differentiation

A) Differentiate both sides of the equation

Unlock document

This preview shows pages 1-2 of the document.
Unlock all 5 pages and 3 million more documents.

Already have an account? Log in

B) Solve for or

dy y′

Example



Determine for y′6x2+y2= 3

Solution:

A) Differentiate

dx (x)

2+y2=d

dx (36)

→ x y2 + 2 dx

dy = 0

B) Solve for dx

y x2dx

dy = − 2

dy =2y

−2x

(cannot go further because do not know what is)= − y

We can also find by using the explicit form of

dy 6x2+y2= 3

Cast y=√36 − x2

(36 )

dy =2

1−x2−2

1(− x)2

= − x

√36 − x2

= − y

Cast y= − √36 − x2

(36 )

dy = − 2

1−x2−2

1(− x)2

√36 − x2

= − y

Example



Find the equation of the tangent line at the point for ,

(√

32 √

32)6x2+y2= 3

Solution:

We have a point, we need a slope

m=dx

dy = − √

√

32= − 1

Point-Slope:

Unlock document

This preview shows pages 1-2 of the document.
Unlock all 5 pages and 3 million more documents.

Already have an account? Log in

Document Summary

Express one variable in terms of the other variable (x) y = f y = 3x2 + e5x y = x3 + 4 + t an2x. The variables are defined in terms of an equation such as: x2 + y2 = 3. 6 ex + ey = 2 lm xy. In some cases it is possible to obtain an explicit expression for one variable in terms of the other variable. Find an explicit expression for x2 + y2 = 3. Y = 36 x2 y in terms of x. Most of the time we cannot obtain an explicit expression from an implicit expression, as in ex + ey = 2 lm xy. Implicit differentiation: differentiate both sides of the equation, solve for dy dx or y . Solution: y for x2 + y2 = 3. 6: differentiate dx (36) d dx (x. 2 + y2 = d dy = 0 y.

Related Questions

help

Using the function y = f(x) obtained in (c), find the .y coordinate corresponding to the value x = -1 / 2 Evaluate both formulas for dy / dx at this point. Why do you think this happens.? Using the explicit derivative formula dy / dx = f (x) obtained in part (c) , find all points on the graph of y = f(x) where the tangent line is horizontal. Then substitute the x and y - coordinates of each such point into the implicit dy / dx formula given in part(b). Again, why does this happen? Again using the explicit derivative formula dy / dx = f(x) from part (c). find all points on the graph of f where the tangent slope equals 6. Now substitute the x- and y- coordinates of each such point into the implicit dy / dx formula given in part (b) to confirm this slope value. It is clear that no point (x, y) for which x + 2y = 0 will appear on the graph of original equation 2x = y - x2 / x +2y. In effect, points on the line x + 2y = 0 are "cut out" of the graph of the function f Relate this observation to the phenomena encountered earlier in parts (e) and (f). Once again referring to the explicit function f(x) obtained in part (c), where does the graph of f have a vertical asymptote? What happens when you substitute this X-value into the original implicit equation (given at the beginning of this worksheet) and attempt to solve for y ? Explain. [ Verification of the implicit derivative given in part (b).] Implicitly differentiate the equation 2x = y - x2 / x = 2y, applying the quotient role on the right-hand side. Clear the denominator and expand terms, then solve for dy / dx and simplify. Note: Solutions to this problem set are t0 be established by traditional algebraic or arithmetic means. using Maple software to verify each result To download the Maple template file prepared to assist you w ith these exercises, go to https;//my,ltu edu and login, select Calculus 1 and click on Workshop. Saw this file to your laptop hard drive and exit the web Hand-generated solutions are to be fully documented In the workshop report including all steps requiring algebraic or calculus computations - merely listing the answers produce with Maple is not sufficient. Printouts of your Maple verifications are required as part of your workshop report. Consider the equation 2x = y -x2 / x+ 2y. Verify that the points .A(1, -1) and B(-2, 4/3) lie on the graph of this equation. In part (j) of this worksheet you will be asked to verify- the derivative formula by implicitly differentiating the equation given above. Evaluate this derivative at points A and B. Solve for y in the original equation, expressing y as an explicit function of x. (For later reference, we will call this function f(x).) Compute dy / dx from this expression and simplify , then evaluate this explicit version of dy / dx at both A aud B. Considering the results of part (b), are these the values you expected? Why or why not? Evaluate each of the two formulas for dy / dx (both implicit and explicit) at the point C(-1, 0). Can you explain your results?

help

Follow the six steps below to solve the following: xy' + 6xy =xe^(5x)

Step 1: REWRITE into standard form (y' + P(x)y + Q(x))

a) How does the given equation differ from standard form?

b) What can be done to put the given equation into standardform?

Step 2: IDENTIFY P(x) and Q(x).

a) Where does the standard form indicate P(x) and Q(x) arelocated?

P(x) =

Q(x) =

Step 3: DETERMINE the integrating factor (e^(?P(x) dx)). PlugP(x) as identified in step 2 into the integrating factor forme^(?P(x) dx) . Hint: Simplify e^(?P(x) dx) if possible

a) What is meant by the integrating factor?

b) If the integrating factor looks like e^(ln (x)) how could itbe simplified?

c) What about e^(1/2 ln (x))?

d) What about e^(-ln (x))?

?P(x) dx =

e^(?P(x) dx) =

Step 4: SUBSTITUTE Q(x) from step 2 and e^(?P(x) dx) from step 3into the solution form ye^(?P(x) dx) = ?Q(x)e^(?P(x) dx) + C.

Step 5: INTEGRATE the right side of the result of step 4. Hint:You may need to simplify and/or rewrite the right side before youattempt to integrate.

Step 6: SOLVE for y. To do this divide both sides by theintegrating factor. Remember that all parts of the right sideincluding the C must be divided by the integrating factor.

Solution is _________________________

ochrerat635

MATH241 Lecture Notes - Lecture 17: Implicit Function

Document Summary

Get access

Related textbook solutions

Calculus

Single Variable Calculus: Early Transcendentals

CALCULUS:EARLY TRANSCENDENTALS

Related Documents

MATH241 Lecture 27: Optimization Problems

MATH 1007 Lecture Notes - Lecture 6: Implicit Function, Product Rule

MATH 111 Lecture Notes - Lecture 12: Implicit Function, Chain Rule

Related Questions