Unit 3. Integration 3A. Differentials, indefinite integration
... a) At sea level the pressure is1 1 kg/cm2 . Solve the equation and find the pressure at the top of Mt. Everest (10 km). b) Find the difference in pressure between the top and bottom of the Green Building. (Pretend it’s 100 meters tall starting at sea level.) Compute the numerical value using a calcu ...
... a) At sea level the pressure is1 1 kg/cm2 . Solve the equation and find the pressure at the top of Mt. Everest (10 km). b) Find the difference in pressure between the top and bottom of the Green Building. (Pretend it’s 100 meters tall starting at sea level.) Compute the numerical value using a calcu ...
Dougherty Lecture 3
... Most texts call (8) the standard form of (7). Solving (8) is the subject of Farlow’s Section 2.1. Note that we divided by a1 (x), which may occasionally be zero. We have not yet discussed the topic of just where we can find a solution, i.e., for which x’s can we solve such an equation. Thus anytime ...
... Most texts call (8) the standard form of (7). Solving (8) is the subject of Farlow’s Section 2.1. Note that we divided by a1 (x), which may occasionally be zero. We have not yet discussed the topic of just where we can find a solution, i.e., for which x’s can we solve such an equation. Thus anytime ...
Parametric Curves
... in the fact that we try to make the curve pass through all of the specified points • A better way is to specify points that control how the curve passes from one point to the next • We do so by specifying a cubic function controlled by four points • The four points are called boundary conditions ...
... in the fact that we try to make the curve pass through all of the specified points • A better way is to specify points that control how the curve passes from one point to the next • We do so by specifying a cubic function controlled by four points • The four points are called boundary conditions ...
1 - JustAnswer
... Crosses the x-axis at -2, 0, and 4; lies above the x-axis between -2 and 0; lies below the x-axis between 0 and 4. ...
... Crosses the x-axis at -2, 0, and 4; lies above the x-axis between -2 and 0; lies below the x-axis between 0 and 4. ...
Fundamental Theorem of Calculus, Riemann Sums, Substitution
... Notice that the integral involves one of the terms above. Substitute the appropriate u. Make sure to change the dx to a du (with relevant factor). Simplify the integral using the appropriate trig identity. Rewrite the new integral in terms of the original non-Ѳ variable (draw a reference right-trian ...
... Notice that the integral involves one of the terms above. Substitute the appropriate u. Make sure to change the dx to a du (with relevant factor). Simplify the integral using the appropriate trig identity. Rewrite the new integral in terms of the original non-Ѳ variable (draw a reference right-trian ...
MATH 645 - HOMEWORK 1: Exercise 1.1.3. Show that
... Proof. By the proof of the triangle inequality and the result on equality from the Cauchy-Schwarz inequality we see that ktf + (1 − t)gk ≤ |t|kf k + |(1 − t)|kgk = 1 where equality occurs if and only if some nontrivial linear combination of f and g is zero. Since these vectors are linearly independe ...
... Proof. By the proof of the triangle inequality and the result on equality from the Cauchy-Schwarz inequality we see that ktf + (1 − t)gk ≤ |t|kf k + |(1 − t)|kgk = 1 where equality occurs if and only if some nontrivial linear combination of f and g is zero. Since these vectors are linearly independe ...
Chapter 4 – Applications of Differentiation
... 4.1 – Maximum and Minimum Values Optimization – the ‘best’ way Absolute maximum/minimum: c is an absolute maximum if f (c) f ( x) x D Local maximum/minimum: c is a local maximum if f (c) f ( x) x some open interval around c The extreme value theorem: If f is continuous on a closed interva ...
... 4.1 – Maximum and Minimum Values Optimization – the ‘best’ way Absolute maximum/minimum: c is an absolute maximum if f (c) f ( x) x D Local maximum/minimum: c is a local maximum if f (c) f ( x) x some open interval around c The extreme value theorem: If f is continuous on a closed interva ...
[Write on board:
... (Analogy: n n+1 for all n N, but we can’t conclude that {n N} is bounded above.) Lemma 7.2.4: If P1 and P2 are any two partitions of [a,b], L(f,P1) U(f,P2). Consequence: The set L = {L(f,P)} = {L(f,P1)}is nonempty, and is bounded above by ANY element of the set U = {U(f,P)} = {U(f,P2)}, so b ...
... (Analogy: n n+1 for all n N, but we can’t conclude that {n N} is bounded above.) Lemma 7.2.4: If P1 and P2 are any two partitions of [a,b], L(f,P1) U(f,P2). Consequence: The set L = {L(f,P)} = {L(f,P1)}is nonempty, and is bounded above by ANY element of the set U = {U(f,P)} = {U(f,P2)}, so b ...
Document
... Most graphing calculators have a built-in numerical differentiation routine that will approximate numerically the values of f (x) for any given value of x. Some graphing calculators have a built-in symbolic differentiation routine that will find an algebraic formula for the derivative, and then eva ...
... Most graphing calculators have a built-in numerical differentiation routine that will approximate numerically the values of f (x) for any given value of x. Some graphing calculators have a built-in symbolic differentiation routine that will find an algebraic formula for the derivative, and then eva ...
Slide
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
PDF English
... equal to 1 divided by square root of 3. Now to find x here, either you can remember your special trig angles and know which values of x make this work. Or you could apply the arc tangent function here. So in either case, the simplest solution here is x equals pi over 6. So if you like, you can draw ...
... equal to 1 divided by square root of 3. Now to find x here, either you can remember your special trig angles and know which values of x make this work. Or you could apply the arc tangent function here. So in either case, the simplest solution here is x equals pi over 6. So if you like, you can draw ...
AP Calculus Multiple Choice: BC Edition – Solutions
... two series combined into one. One of the series is ...
... two series combined into one. One of the series is ...
Newton and Leibniz: the Calculus Controversy
... Using infinite series and the already established power rule for integrals, Newton was able to calculate the areas under curves that others previously could not. He was also able to calculate the tangent lines to these curves. He called his calculus the “method of fluxions” and he thought of everyth ...
... Using infinite series and the already established power rule for integrals, Newton was able to calculate the areas under curves that others previously could not. He was also able to calculate the tangent lines to these curves. He called his calculus the “method of fluxions” and he thought of everyth ...
With(out) A Trace - Matrix Derivatives the Easy Way
... Calculus Property 3: Chain Rule Let: z = 2x21 + x1 cos(x2 ) Then, by definition: ...
... Calculus Property 3: Chain Rule Let: z = 2x21 + x1 cos(x2 ) Then, by definition: ...
word
... Although the order in which you try various methods is not fixed, I find the following helpful. There is no substitution for practice when it comes to quickly identifying the proper method. 1. Can the integrand first be simplified using basic algebra, logarithm properties or trigonometric identities ...
... Although the order in which you try various methods is not fixed, I find the following helpful. There is no substitution for practice when it comes to quickly identifying the proper method. 1. Can the integrand first be simplified using basic algebra, logarithm properties or trigonometric identities ...
f ` (x)
... Problem of the Day Let f and g be odd functions. If p, r, and s are nonzero functions defined as follows, which must be odd? I. p(x) = f(g(x)) II. r(x) = f(x) + g(x) III. s(x) = f(x)g(x) A) I B) II ...
... Problem of the Day Let f and g be odd functions. If p, r, and s are nonzero functions defined as follows, which must be odd? I. p(x) = f(g(x)) II. r(x) = f(x) + g(x) III. s(x) = f(x)g(x) A) I B) II ...
2-6 Special_Functions 9-9
... The right portion of the graph is the constant function f(x) = 1. The constant function is defined for {x | x ≥ 2}. Answer: ...
... The right portion of the graph is the constant function f(x) = 1. The constant function is defined for {x | x ≥ 2}. Answer: ...
2.4 Continuity
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
Differentiation - Trig, Log and Exponential
... The gradient of sin x° is equal to zero when x = 90° and 270°, and so is cos x°. But when we come to measure the gradient at x = 0°, we obtain a strange-looking value of 0.0175, whereas cos 0° = 1. The measured gradient is therefore about 57 times too small. Similarly, the gradient at x = 180° comes ...
... The gradient of sin x° is equal to zero when x = 90° and 270°, and so is cos x°. But when we come to measure the gradient at x = 0°, we obtain a strange-looking value of 0.0175, whereas cos 0° = 1. The measured gradient is therefore about 57 times too small. Similarly, the gradient at x = 180° comes ...
College Algebra Chapter 2 Functions and Graphs
... A new job offer in sales promises a base salary of $3000 a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewisedefined function (in dollars) to model the expected total monthly salary as a func ...
... A new job offer in sales promises a base salary of $3000 a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewisedefined function (in dollars) to model the expected total monthly salary as a func ...
What is the Second Fundamental Theorem of Calculus
... Objectives: to see that a function can be defined by an integral and that it can be differentiated to find the maximum or minimum. Grouping: students are given opportunity to work in cooperative setting during this class. Time is purposely set aside for students to work collaboratively on at least o ...
... Objectives: to see that a function can be defined by an integral and that it can be differentiated to find the maximum or minimum. Grouping: students are given opportunity to work in cooperative setting during this class. Time is purposely set aside for students to work collaboratively on at least o ...
