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Slide 1: Review for Midterm I Math 1a October 21, 2007 Announcements Midterm I 10/24, Hall 7-9pm, Hall A and D Old exams and solutions on website problem sessions every night, extra MQC hours

Slide 2: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 3: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 4: The concept of Limit Learning Objectives state the informal definition of a limit (two- and one-sided) observe limits on a graph guess limits by algebraic manipulation guess limits by numerical information

Slide 5: Heuristic Definition of a Limit Definition We write lim f (x) = L x→a and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a.

Slide 7: The error-tolerance game L a

Slide 8: The error-tolerance game L a

Slide 9: The error-tolerance game L a

Slide 10: The error-tolerance game This tolerance is too big L a

Slide 11: The error-tolerance game L a

Slide 12: The error-tolerance game Still too big L a

Slide 13: The error-tolerance game L a

Slide 14: The error-tolerance game This looks good L a

Slide 15: The error-tolerance game So does this L a

Slide 16: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 17: Computation of Limits Learning Objectives know basic limits like limx→a x = a and limx→a c = c use the limit laws to compute elementary limits use algebra to simplify limits use the Squeeze Theorem to show a limit

Slide 18: Limit Laws Suppose that c is a constant and the limits lim f (x) and lim g (x) x→a x→a exist. Then 1. lim [f (x) + g (x)] = lim f (x) + lim g (x) x→a x→a x→a 2. lim [f (x) − g (x)] = lim f (x) − lim g (x) x→a x→a x→a 3. lim [cf (x)] = c lim f (x) x→a x→a 4. lim [f (x)g (x)] = lim f (x) · lim g (x) x→a x→a x→a

Slide 19: Limit Laws, continued lim f (x) f (x) = x→a 5. lim , if lim g (x) = 0. x→a g (x) lim g (x) x→a x→a n n 6. lim [f (x)] = lim f (x) (follows from 3 repeatedly) x→a x→a 7. lim c = c x→a 8. lim x = a x→a 9. lim x n = an (follows from 6 and 8) x→a √ √ 10. lim n x = n a x→a n 11. lim f (x) = lim f (x) (If n is even, we must additionally n x→a x→a assume that lim f (x) > 0) x→a

Slide 20: Direct Substitution Property Theorem (The Direct Substitution Property) If f is a polynomial or a rational function and a is in the domain of f , then lim f (x) = f (a) x→a

Slide 22: Theorem (The Squeeze/Sandwich/Pinching Theorem) If f (x) ≤ g (x) ≤ h(x) when x is near a (as usual, except possibly at a), and lim f (x) = lim h(x) = L, x→a x→a then lim g (x) = L. x→a

Slide 25: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 26: Limits involving infinity Learning Objectives know vertical asymptotes and limits at the discontinuities of ”famous” functions intuit limits at infinity by eyeballing the expression show limits at infinity by algebraic manipulation

Slide 27: Definition Let f be a function defined on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x sufficiently large. Definition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x→∞ x→−∞ y = L is a horizontal line!

Slide 28: Theorem Let n be a positive integer. Then 1 limx→∞ =0 xn limx→−∞ x1n =0

Slide 29: Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C0 D∞

Slide 30: Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C0 D∞

Slide 31: Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) =3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When finding limits of algebraic expressions at infinitely, look at the highest degree terms.

Slide 32: Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) =3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When finding limits of algebraic expressions at infinitely, look at the highest degree terms.

Slide 33: Infinite Limits Definition The notation lim f (x) = ∞ x→a means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. Definition The notation lim f (x) = −∞ x→a means that the values of f (x) can be made arbitrarily large negative (as large as we please) by taking x sufficiently close to a but not equal to a. Of course we have definitions for left- and right-hand infinite limits.

Slide 34: Vertical Asymptotes Definition The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following is true: limx→a f (x) = ∞ limx→a f (x) = −∞ limx→a+ f (x) = ∞ limx→a+ f (x) = −∞ limx→a− f (x) = ∞ limx→a− f (x) = −∞

Slide 35: Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find limt→a− f (t) and limt→a+ f (t) for each a at which f is not continuous.

Slide 36: Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find limt→a− f (t) and limt→a+ f (t) for each a at which f is not continuous. Solution The denominator factors as (t − 1)(t − 2). We can record the signs of the factors on the number line.

Slide 37: − + 0 (t − 1) 1

Slide 38: − + 0 (t − 1) 1 − + 0 (t − 2) 2

Slide 39: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2)

Slide 40: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) f (t) 1 2

Slide 41: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) + f (t) 1 2

Slide 42: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ + f (t) 1 2

Slide 43: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − + f (t) 1 2

Slide 44: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − ∞ + f (t) 1 2

Slide 45: − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − ∞ + + f (t) 1 2

Slide 49: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 50: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 51: Continuity Learning Objectives intuitive notion of continuity definition of continuity at a point and on an interval ways a function can fail to be continuous at a point

Slide 52: Definition of Continuity Definition Let f be a function defined near a. We say that f is continuous at a if lim f (x) = f (a). x→a

Slide 54: Free Theorems Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (−∞, ∞). (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Slide 55: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 56: The Limit Laws give Continuity Laws Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. f + g 2. f − g 3. cf 4. fg f 5. (if g (a) = 0) g

Slide 57: Transcendental functions are continuous, too Theorem The following functions are continuous wherever they are defined: 1. sin, cos, tan, cot sec, csc 2. x → ax , loga , ln 3. sin−1 , tan−1 , sec−1

Slide 60: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 61: The Intermediate Value Theorem Learning Objectives state IVT use IVT to show that a function takes a certain value use IVT to show that a certain equation has a solution reason with IVT

Slide 62: A Big Time Theorem Theorem (The Intermediate Value Theorem) Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N.

Slide 63: Illustrating the IVT f (x) x

Slide 64: Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) x

Slide 65: Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) f (b) f (a) x a b

Slide 66: Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). f (x) f (b) N f (a) x a b

Slide 67: Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a c b

Slide 68: Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a b

Slide 69: Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] and let N be any number between f (a) and f (b), where f (a) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a c1 c2 c3 b

Slide 70: Using the IVT Example Prove that the square root of two exists. Proof. Let f (x) = x 2 , a continuous function on [1, 2]. Note f (1) = 1 and f (2) = 4. Since 2 is between 1 and 4, there exists a point c in (1, 2) such that f (c) = c 2 = 2.

Slide 71: True or False At one point in your life your height in inches equaled your weight in pounds.

Slide 73: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 74: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 75: Concept Learning Objectives state the definition of the derivative Given the formula for a function, find its derivative at a point “from scratch,” i.e., using the definition Given numerical data for a function, estimate its derivative at a point. given the formula for a function and a point on the graph of the function, find the (slope of, equation for) the tangent line

Slide 76: The definition Definition Let f be a function and a a point in the domain of f . If the limit f (a + h) − f (a) f (a) = lim h h→0 exists, the function is said to be differentiable at a and f (a) is the derivative of f at a.

Slide 77: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 78: The Derivative as a function Learning Objectives given a function, find the derivative of that function from scratch and give the domain of f’ given a function, find its second derivative given the graph of a function, sketch the graph of its derivative

Slide 79: Derivatives Theorem If f is differentiable at a, then f is continuous at a.

Slide 80: How can a function fail to be continuous?

Slide 82: The second derivative If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change!

Slide 84: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 85: Implications of the derivative Learning objectives Given the graph of the derivative of a function... determine where the function is increasing and decreasing determine where the function is concave up and concave down sketch the graph of the original function find and interpret inflection points

Slide 87: Fact If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is decreasing on (a, b), then f (x) ≤ 0 for all x in (a, b). Fact If f (x) > 0 for all x in (a, b), then f is increasing on (a, b). If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).

Slide 88: Definition A function is called concave up on an interval if f is increasing on that interval. A function is called concave down on an interval if f is decreasing on that interval.

Slide 89: Fact If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is concave down on (a, b), then f (x) ≤ 0 for all x in (a, b). Fact If f (x) > 0 for all x in (a, b), then f is concave up on (a, b). If f (x) < 0 for all x in (a, b), then f is concave down on (a, b).

Slide 90: Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation

Slide 91: Computing Derivatives Learning Objectives the power rule the constant multiple rule the sum rule the difference rule derivative of x → e x is e x (by definition of e)

Slide 92: Theorem (The Power Rule) Let r be a real number. Then dr x = rx r −1 dx

Slide 93: Rules for Differentiation Theorem Let f and g be differentiable functions at a, and c a constant. Then (f + g ) (a) = f (a) + g (a) (cf ) (a) = cf (a)

Slide 94: Rules for Differentiation Theorem Let f and g be differentiable functions at a, and c a constant. Then (f + g ) (a) = f (a) + g (a) (cf ) (a) = cf (a) It follows that we can differentiate all polynomials.

Slide 95: Derivatives of exponential functions Fact dx = ex dx e