Here
... Now, if g were differentiable, then for an arbitrary unit vector v we would have Dv f (0, 0) = v1 ∂1 f + v2 ∂2 f = 0 by the chain rule. But Dv f = g(v), so g(v) must be zero. ...
... Now, if g were differentiable, then for an arbitrary unit vector v we would have Dv f (0, 0) = v1 ∂1 f + v2 ∂2 f = 0 by the chain rule. But Dv f = g(v), so g(v) must be zero. ...
Derivatives - Pauls Online Math Notes
... We have θ ′ = 0.01 rad/min. and want to find x′ . We can use various trig fcns but easiest is, x x′ sec θ = ⇒ sec θ tan θ θ ′ = ...
... We have θ ′ = 0.01 rad/min. and want to find x′ . We can use various trig fcns but easiest is, x x′ sec θ = ⇒ sec θ tan θ θ ′ = ...
Linear approximation and the rules of differentiation
... Linear approximation and the rules of differentiation The whole body of calculus rest on the idea that if you break things up into little pieces, you can approximate them linearly. Geometrically speaking, we approximate an arc of a curve by a tangent line segment or, in higher dimensions, a piece of ...
... Linear approximation and the rules of differentiation The whole body of calculus rest on the idea that if you break things up into little pieces, you can approximate them linearly. Geometrically speaking, we approximate an arc of a curve by a tangent line segment or, in higher dimensions, a piece of ...
x – 1 - Webs
... So, in using the Product Rule, • There must be a multiplication of terms or expressions which will result in a product (Product Rule) • We have to always make sure that we have two terms (first term = u and second term = v) • If we do not have two terms, we have to chunk the number of terms that we ...
... So, in using the Product Rule, • There must be a multiplication of terms or expressions which will result in a product (Product Rule) • We have to always make sure that we have two terms (first term = u and second term = v) • If we do not have two terms, we have to chunk the number of terms that we ...
a review sheet for test #02
... Section 2.9: The Mean Value Theorem Theorem 9.1 (Rolle’s Theorem): (The “What goes up must come down” Theorem) If f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), and f(a) = f(b), then there is a number c (a, b) such that f (c) = 0. Theorem 9.2 (Corollary #1 to R ...
... Section 2.9: The Mean Value Theorem Theorem 9.1 (Rolle’s Theorem): (The “What goes up must come down” Theorem) If f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), and f(a) = f(b), then there is a number c (a, b) such that f (c) = 0. Theorem 9.2 (Corollary #1 to R ...
Here
... (vi) What does a non-differentiable function look like, in terms of its graph? Give examples. (b) 11.2 Derivative rules. (i) Find the derivative of x3 using the definition of the derivative and using the power rule. Check that they agree. (ii) If f 0 (2) = 4 and g 0 (2) = −1, what is (4f − 2g)0 (2)? ...
... (vi) What does a non-differentiable function look like, in terms of its graph? Give examples. (b) 11.2 Derivative rules. (i) Find the derivative of x3 using the definition of the derivative and using the power rule. Check that they agree. (ii) If f 0 (2) = 4 and g 0 (2) = −1, what is (4f − 2g)0 (2)? ...
Practical Guide to Derivation
... This follows from the question of how to derive the function y = xsin(5x). One could not take such a derivative without the above consideration. It would go as follows. xsin(5x) 1sin(5x) + x(5cos(5x)) dx Consider why this might be so. Like the expansion of (x+5)2 = x2+10x+25, the middle term must be ...
... This follows from the question of how to derive the function y = xsin(5x). One could not take such a derivative without the above consideration. It would go as follows. xsin(5x) 1sin(5x) + x(5cos(5x)) dx Consider why this might be so. Like the expansion of (x+5)2 = x2+10x+25, the middle term must be ...
Math 171 Final Exam Review: Things to Know
... − Find critical points. − Extreme Value Theorem (page 238) − Know how to locate absolute extrema on a closed interval. (See page 241) • Test for Intervals of Increase and Decrease (page 246) • First Derivative Test (page 249) • Concavity and Inflection Points − See the test for concavity on page 252 ...
... − Find critical points. − Extreme Value Theorem (page 238) − Know how to locate absolute extrema on a closed interval. (See page 241) • Test for Intervals of Increase and Decrease (page 246) • First Derivative Test (page 249) • Concavity and Inflection Points − See the test for concavity on page 252 ...
Partial solutions to Even numbered problems in
... 32. This problem is tricky. We have to apply the MVT to the velocity function (first derivative), so that we can make a conclusion about acceleration (second derivative). Let p(t) be the position function of the car at time t. Then p0 is the velocity function of the car. Also, p0 is continuous and d ...
... 32. This problem is tricky. We have to apply the MVT to the velocity function (first derivative), so that we can make a conclusion about acceleration (second derivative). Let p(t) be the position function of the car at time t. Then p0 is the velocity function of the car. Also, p0 is continuous and d ...
1 Lecture 4 - Integration by parts
... Mathematically speaking, this is imprecise. After all, what, exactly, is the mathematical meaning of “infinitesimal”? Maybe a number that is really really really really really small?? Maybe some kind of number smaller than all positive numbers but bigger that 0?? Or is and ‘infinitesimal’ really a n ...
... Mathematically speaking, this is imprecise. After all, what, exactly, is the mathematical meaning of “infinitesimal”? Maybe a number that is really really really really really small?? Maybe some kind of number smaller than all positive numbers but bigger that 0?? Or is and ‘infinitesimal’ really a n ...
Final Review - Mathematical and Statistical Sciences
... Computing derivatives: recognizing when the product rule, quotient rule and chain rule should be used, correctly applying them (even multiple times for the same function). Computing derivatives ...
... Computing derivatives: recognizing when the product rule, quotient rule and chain rule should be used, correctly applying them (even multiple times for the same function). Computing derivatives ...
Lesson 1-1 - Louisburg USD 416
... Ex. 5 Find a function y = f(x) whose derivative dy/dx = tan x and satisfies the condition f(3) = 5. ...
... Ex. 5 Find a function y = f(x) whose derivative dy/dx = tan x and satisfies the condition f(3) = 5. ...
Math 163 Notes Section 5.3
... The first derivative of a function represents the rate of change of the function. The second derivative, then, represents the rate of change of the first derivative—it indicates how fast the function is increasing or decreasing. If st represents position at time t, then o vt s' t gives ...
... The first derivative of a function represents the rate of change of the function. The second derivative, then, represents the rate of change of the first derivative—it indicates how fast the function is increasing or decreasing. If st represents position at time t, then o vt s' t gives ...
Course Narrative
... • Corresponding characteristics of the graphs of ƒ, ƒ’ and ƒ” • Relationship between the concavity of ƒ and the sign of ƒ” • Points of inflection as places where concavity changes Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity • Optimization, bo ...
... • Corresponding characteristics of the graphs of ƒ, ƒ’ and ƒ” • Relationship between the concavity of ƒ and the sign of ƒ” • Points of inflection as places where concavity changes Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity • Optimization, bo ...
2.2 Derivative of Polynomial Functions A Power Rule Consider the
... Calculus and Vectors – How to get an A+ ...
... Calculus and Vectors – How to get an A+ ...
Calc I Review Sheet
... Note: It follows by the chain rule that if u(x) is differentiable on [a, b] then d dx ...
... Note: It follows by the chain rule that if u(x) is differentiable on [a, b] then d dx ...
Math 108, Final Exam Checklist
... • Using IVT to find where a function is positive and where it is negative. • Definition of derivative • Tangent line Chapter 3: Differentiation rules • Differentiation rules: sum, product, quotient and chain rule. • Derivatives of elementary functions: E.g. derivative of xa vs ax • Implicit differen ...
... • Using IVT to find where a function is positive and where it is negative. • Definition of derivative • Tangent line Chapter 3: Differentiation rules • Differentiation rules: sum, product, quotient and chain rule. • Derivatives of elementary functions: E.g. derivative of xa vs ax • Implicit differen ...
Final Exam topics - University of Arizona Math
... is close to L and suppose that f(x) can be made as close as we want to L by making x larger. Then we say that the limit of f(x) as x approaches infinity is L and we write Vertical Asymptote Let f be a function which is defined on some open interval containing “a” except possibly at x = a. We write i ...
... is close to L and suppose that f(x) can be made as close as we want to L by making x larger. Then we say that the limit of f(x) as x approaches infinity is L and we write Vertical Asymptote Let f be a function which is defined on some open interval containing “a” except possibly at x = a. We write i ...
![The Fundamental Theorem of Calculus [1]](http://s1.studyres.com/store/data/020099492_1-4a7fbd2304ff84025ef2f0bc4ff924ca-300x300.png)
The Fundamental Theorem of Calculus [1]
... when f is continuous. Roughly speaking, 2.3 says that if we first integrate f and then differentiate the result, we get back to the original function f . This shows that an antiderivative can be reversed by a differentiation, and it also guarantees the existence, continuity, differentiability of antider ...
... when f is continuous. Roughly speaking, 2.3 says that if we first integrate f and then differentiate the result, we get back to the original function f . This shows that an antiderivative can be reversed by a differentiation, and it also guarantees the existence, continuity, differentiability of antider ...
Microsoft Word Format
... Limit: if f(x) becomes arbitrarily close to L as x approaches C from either side, then the limit of f(x) as x approaches C is L lim f(x) = L x Limits that fail to exist: 1. f(x) approaches a different number from the right side of C than from the left side of C 2. f(x) increases or decreases witho ...
... Limit: if f(x) becomes arbitrarily close to L as x approaches C from either side, then the limit of f(x) as x approaches C is L lim f(x) = L x Limits that fail to exist: 1. f(x) approaches a different number from the right side of C than from the left side of C 2. f(x) increases or decreases witho ...
LESSON 21 - Math @ Purdue
... Note 1. We call dx and dy differentials. By the nature of derivatives (in particular, because we would assume that ∆x → 0), the smaller ∆x is, the better the approximation of ∆y. Think of ∆ as the actual change and d as the infinitesimal change. This is why we use dx in an integral but not ∆x because ...
... Note 1. We call dx and dy differentials. By the nature of derivatives (in particular, because we would assume that ∆x → 0), the smaller ∆x is, the better the approximation of ∆y. Think of ∆ as the actual change and d as the infinitesimal change. This is why we use dx in an integral but not ∆x because ...
Calculus I Midterm II Review Materials Solutions to the practice
... Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the answer. 1, Definition of Derivative, Differentials? Relation with change rate, tangent, velocity? Express a limit as √ a derivative and evaluate. ...
... Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the answer. 1, Definition of Derivative, Differentials? Relation with change rate, tangent, velocity? Express a limit as √ a derivative and evaluate. ...
MATH M25A - Moorpark College
... Evaluate the limit of a function using numerical and algebraic techniques, the properties of limits, and analysis techniques. Evaluate one-sided and two-sided limits for algebraic and trigonometric functions. Determine analytically whether a limit fails to exist. Determine whether a function is cont ...
... Evaluate the limit of a function using numerical and algebraic techniques, the properties of limits, and analysis techniques. Evaluate one-sided and two-sided limits for algebraic and trigonometric functions. Determine analytically whether a limit fails to exist. Determine whether a function is cont ...
3.3 Derivatives of Logarithmic and Exponential Functions (10/21
... In this section we will be using the product rule, quotient rule, and chain rule to differentiate functions, but our functions will involve exponentials and logarithms, so we need to discuss their derivatives. The proofs of these can be found in your book. First, let's review the rules from chap ...
... In this section we will be using the product rule, quotient rule, and chain rule to differentiate functions, but our functions will involve exponentials and logarithms, so we need to discuss their derivatives. The proofs of these can be found in your book. First, let's review the rules from chap ...