Math 111 – Calculus I
... The following is a fundamental result involving absolute maximums and minimums of continuous functions. It is called the Extreme Value Theorem. The formal statement of the theorem follows. Theorem 9.5(the Extreme Value Theorem): Assume f is a continuous function on a closed interval [a,b]. Then, f a ...
... The following is a fundamental result involving absolute maximums and minimums of continuous functions. It is called the Extreme Value Theorem. The formal statement of the theorem follows. Theorem 9.5(the Extreme Value Theorem): Assume f is a continuous function on a closed interval [a,b]. Then, f a ...
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... every continuous function has a well defined tangent - at least at “almost all” points. As the Weierstrass function shows that this is clearly not the case. The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [?]. Alternative examples of continuo ...
... every continuous function has a well defined tangent - at least at “almost all” points. As the Weierstrass function shows that this is clearly not the case. The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [?]. Alternative examples of continuo ...
Unit 4 Rates of Change Introduction
... worked independently on developing calculus. However, in a dispute that continued until the end of their lives, each claimed that the other stole had stolen his work. Here we concentrate on the application of finding the slope of graphs to determine the speed and acceleration of moving objects. Dist ...
... worked independently on developing calculus. However, in a dispute that continued until the end of their lives, each claimed that the other stole had stolen his work. Here we concentrate on the application of finding the slope of graphs to determine the speed and acceleration of moving objects. Dist ...
Lecture 20: Further graphing
... • Classify the discontinuities. This amounts to computing limits – if you find that the function is discontinuous at a point, find the limits on the left and right. – If both one-sided limits exist and are equal, this discontinuity is removable. – If both one-sides limits exist but are different, t ...
... • Classify the discontinuities. This amounts to computing limits – if you find that the function is discontinuous at a point, find the limits on the left and right. – If both one-sided limits exist and are equal, this discontinuity is removable. – If both one-sides limits exist but are different, t ...
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... If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can therefore be regarded as a secant vector. If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the same direction as r(t + h) – r(t). As h 0, it appears that this vector approac ...
... If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can therefore be regarded as a secant vector. If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the same direction as r(t + h) – r(t). As h 0, it appears that this vector approac ...
Quiz Three Calculus A Professor D. Olles Name - RIT
... The graph of the function does not approach any ONE y-value, in particular, as x approaches 3. From the left, the function approaches 1 and from the right, the function approaches 4. ...
... The graph of the function does not approach any ONE y-value, in particular, as x approaches 3. From the left, the function approaches 1 and from the right, the function approaches 4. ...
Calculus I Homework: Linear Approximation and Differentials Page
... The blue line is the tangent line L(x), the red line is the function g(x), and the dots are where we evaluated to estimate the two numbers. We are evaluating along the tangent line rather than along the function g(x). We do this because it is easier to compute a numerical value along the tangent lin ...
... The blue line is the tangent line L(x), the red line is the function g(x), and the dots are where we evaluated to estimate the two numbers. We are evaluating along the tangent line rather than along the function g(x). We do this because it is easier to compute a numerical value along the tangent lin ...
MATH141 – Tutorial 2
... Solution. The domain on which the parabola is negative is (−∞, 0) ∪ (1, ∞); we use linearity to break the integral into three regions. Note that the integrand will be the same, although the absolute value guarantees that the function stays positive. Z 0 ...
... Solution. The domain on which the parabola is negative is (−∞, 0) ∪ (1, ∞); we use linearity to break the integral into three regions. Note that the integrand will be the same, although the absolute value guarantees that the function stays positive. Z 0 ...
On the number e, its irrationality, and factorials
... number of times. Therefore there can be no integers a and b for which 2 = a/b. All proofs that a number r is irrational follow this pattern of logic, called proof by contradiction: To prove that r is irrational we assume r = a/b for some integers a and b and then show (somehow) that this assumption ...
... number of times. Therefore there can be no integers a and b for which 2 = a/b. All proofs that a number r is irrational follow this pattern of logic, called proof by contradiction: To prove that r is irrational we assume r = a/b for some integers a and b and then show (somehow) that this assumption ...
Algebra 2 – PreAP/GT
... A student group is selling chocolate bars for $2 each. The function f x 2 x gives the amount of money collected after selling x chocolate bars. f x would be considered a discrete function since only whole number of chocolate bars can be sold which would result in a graph of separated point ...
... A student group is selling chocolate bars for $2 each. The function f x 2 x gives the amount of money collected after selling x chocolate bars. f x would be considered a discrete function since only whole number of chocolate bars can be sold which would result in a graph of separated point ...
Though the universal quantifier distributes over conjunction, it does
... Technique: Since the assumption asserts the distributional properties of the universal quantifier for all possible instantiations of p(X) and q(X), finding a single example for which the assumption does not hold is sufficient to prove the theorem. Let the domain be Z, the set of integers. Let tval(S ...
... Technique: Since the assumption asserts the distributional properties of the universal quantifier for all possible instantiations of p(X) and q(X), finding a single example for which the assumption does not hold is sufficient to prove the theorem. Let the domain be Z, the set of integers. Let tval(S ...
log
... 7. The function is continuous, increasing, and one-to-one. 8. The graph is concave downward. D. Examples 1. By comparing areas, show that ln2 <1. ...
... 7. The function is continuous, increasing, and one-to-one. 8. The graph is concave downward. D. Examples 1. By comparing areas, show that ln2 <1. ...
SUBJECTS OF THE FINAL EXAMINATION THE SUBJECTS OF THE
... • SUGGESTIONS FOR FURTHER STUDY IN CALCULUS AND ANALYSIS. This term we have seen one variable calculus: continuity, limits, derivative and integration of realvalued functions of a real variable. We have not given the proofs of all the properties of those. For a rigorous presentation of all these, I ...
... • SUGGESTIONS FOR FURTHER STUDY IN CALCULUS AND ANALYSIS. This term we have seen one variable calculus: continuity, limits, derivative and integration of realvalued functions of a real variable. We have not given the proofs of all the properties of those. For a rigorous presentation of all these, I ...
Challenge #10 (Arc Length)
... You have probably noticed that there seem to be only a few functions whose arc lengths we can actually find. For example, we cannot find the length of an arc on the simplest functions like y x 2 , y 1x , y e x , y sin x since we cannot find an antiderivative for the integrand ...
... You have probably noticed that there seem to be only a few functions whose arc lengths we can actually find. For example, we cannot find the length of an arc on the simplest functions like y x 2 , y 1x , y e x , y sin x since we cannot find an antiderivative for the integrand ...
CHAP08 Integration - Faculty of Science and Engineering
... Differential Calculus is all about slopes and Integral Calculus is all about areas and you might not think that slopes and areas have much to do with each other, apart from being different aspects of a graph. But the surprising fact is that these are inverse operations. An example of a pair of inver ...
... Differential Calculus is all about slopes and Integral Calculus is all about areas and you might not think that slopes and areas have much to do with each other, apart from being different aspects of a graph. But the surprising fact is that these are inverse operations. An example of a pair of inver ...
Final review
... b) A general rule based on the above illustrations is: "The integral of any function with ______ symmetry over limits that are ___________ with respect to the y-axis is always identically ________." c) For each integral below: i) make a sketch of the integrand, ii) shade in the total area represente ...
... b) A general rule based on the above illustrations is: "The integral of any function with ______ symmetry over limits that are ___________ with respect to the y-axis is always identically ________." c) For each integral below: i) make a sketch of the integrand, ii) shade in the total area represente ...
4.2 Mean Value Theorem
... Comments. We “know” that the derivative of a function tells us about the function itself right? Well, how do we know this? Suppose that f 0 (x) = 0 for all values of x. Is it obvious what this means about f (x)? Suppose if f 0 (x) > 0 for all x in an interval. Is it obvious what this means about f ( ...
... Comments. We “know” that the derivative of a function tells us about the function itself right? Well, how do we know this? Suppose that f 0 (x) = 0 for all values of x. Is it obvious what this means about f (x)? Suppose if f 0 (x) > 0 for all x in an interval. Is it obvious what this means about f ( ...
C1 : ALGEBRA AND FUNCTIONS 1
... know y=ln x is the inverse function of y=ex know the properties of the function ln x and its graph know the properties of the function ex and its graph can solve equations involving y=lnx can solve equations involving y= ex ...
... know y=ln x is the inverse function of y=ex know the properties of the function ln x and its graph know the properties of the function ex and its graph can solve equations involving y=lnx can solve equations involving y= ex ...
Derivatives and Integrals of Vector Functions
... Solution: (a) According to Theorem 2, we differentiate each component of r: r ′(t) = 3t2i + (1 – t)e–t j + 2 cos 2t k ...
... Solution: (a) According to Theorem 2, we differentiate each component of r: r ′(t) = 3t2i + (1 – t)e–t j + 2 cos 2t k ...
Practice Test III
... life insurance. Use a graphing utility to fit an exponential function to the data. Predict the annual premium for a 70 year old woman. (Hint after using your calculator, write your final equation in the form of ...
... life insurance. Use a graphing utility to fit an exponential function to the data. Predict the annual premium for a 70 year old woman. (Hint after using your calculator, write your final equation in the form of ...
Calculus - maccalc
... 6. To what new value should f(1) be changed to make f continuous at x = 1? ______ 7. What is the domain of this function? (Interval Notation) ________________________ 8. What is the range of this function? (Interval Notation) ______________________ ...
... 6. To what new value should f(1) be changed to make f continuous at x = 1? ______ 7. What is the domain of this function? (Interval Notation) ________________________ 8. What is the range of this function? (Interval Notation) ______________________ ...