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Barry McQuarrie’s I Glossary & Technique Quiz

Instructions: For each (groups are separated by horizontal lines), match or quantity in left column to de- scriptions that from the right column. There may be more than one match that is possible, and you might not use all the items from the right column in each group.

1. The quantity x = a is called this if lim f(x) = ±∞. Local Maximum x→a 2. A vertical or horizontal on a graph which a approaches. 3. Measures how a function bends, or in other words, the func- Vertical Asymptote tion’s . 4. If lim f(x) = L, then y = L is called this. Concave Up x→±∞ 5. If the function f(x) is above its line at x = a, then Horizontal Asymptote the function has this property at x = a. The function has this property if f 00(a) > 0. of Inflection 6. The function f(x) could have this property if f 0(a) = 0. 7. Occurs if the function changes concavity at x = a. This is Concavity possible if f 00(a) = 0.

Concave Down 8. If the function f(x) is below its tangent line at x = a, then the function has this property at x = a. The function has this property if f 00(a) < 0. Absolute Maximum 9. Any of the maximum or minimum values for a function. Extrema 10. The largest value the function takes over its entire domain.

1. This quantity involves a , usually something like Antidifferentiation g(x) + C (although other forms are possible), and when you assign different values to the constant C you get different dy/dx . 2. The process of finding the derivative of a function f(x). 3. A family of curves.

Differentiation 4. The process of finding an antiderivative of a function f(x). 5. Informally, this quantity can be written as dx and represents a small amount of x. 6. This quantity is included when an antidifferentiation is per- Differential formed. 7. This quantity represents the instantaneous of change Family of Curves of y with respect to the x.

Page 1 of 4 Barry McQuarrie’s Calculus I Glossary & Technique Quiz

1. ex + ey 2. exey x+y e = 3. (x + y)ex+y−1 4. x − y ln(ex − ey) = 5. xy ln(ex−y) = x 6. y 7. ln(ex − ey)

1. This quantity is always non-negative. If v(t) is the , Z t2 this quantity is given by |v(t)| dt. t1 g(x + h) − g(x) Indeterminant Form 2. lim h→0 h Differentiation 3. How far a particle has moved during a from t1 to t2. If v(t) is the velocity, this quantity is given by Z t2 Traveled v(t) dt. t1

Integration 4. Mathematically easy. 5. Mathematically hard.

x − x0 x1 − x0 of tangent line to a at a 1. = y − y0 y1 − y0 point 2. The derivative of the function.

Equation of a straight line 3. The derivative of the function evaluated at the point. 4. z = f(x, y) in R2 5. f(x, y) = 0

Explicit function in R2 6. y = f(x)

7. y − y0 = m(x − x0) in R3 8. x = f(t), y = g(t), t > 0 3 curve in R 9. x = f(t), y = g(t), z = h(t), t > 0

1. − cos(x3) d sin(x3) = dx 2. cos(x3)

2 3 Z 3. −3x cos(x ) 3 x2 sin(x3) dx = 4. 3x2 cos(x3)

Page 2 of 4 Barry McQuarrie’s Calculus I Glossary & Technique Quiz

Z 1. Formally, this quantity looks like f(x) dx. When evalu- Integration ated it yields a family of curves as the solution. Definite 2. The process of evaluating an integral (definite, indefinite, or improper) of a function f(x). Indefinite Integral Z b 3. Formally, this quantity looks like f(x) dx where the in- a tegrand f(x) is infinite for some x ∈ [a, b], or a → −∞ or b → ∞. Integral Z b 4. For f(x) dx, this quantity is f(x). a Limits of Integration 5. Used as a way to refer to any of the specific types of . Z b Integrand 6. For f(x) dx, this quantity is a and b. a d Z x Z b FTC Part 1: w(t) dt = 7. Formally, this quantity looks like f(x) dx. When evalu- dx a a ated it yields a number. b Z d 8. w(x) FTC Part 2: [w(x)] dx = dx a 9. w(b) − w(a)

1. f 0(g(x)) 2. f(g0(x)) 3. f 0(g(x))g0(x) 4. f 0(g0(x))g0(x) g(x)f 0(x) − f(x)g0(x) 5. d f(x) (g(x))2 = dx g(x) g(x)f 0(x) − f(x)g0(x) 6. g(x) d [f(x)g(x)] = g0(x)f 0(x) − f(x)g(x) dx 7. (g(x))2 d g(x)f 0(x) + f(x)g0(x) [f(g(x))] = 8. dx (g(x))2 9. f 0(x)g0(x) 10. f 0(x)g(x) + f(x)g0(x) 11. f(x)g(x) 12. g(x)

Page 3 of 4 Barry McQuarrie’s Calculus I Glossary & Technique Quiz

d 1. Could be used to evaluate [xx]. dx 2. Involves finding relationships between rates of changes. 3. Could be used to find dy/dx if sin(xy) = cos(xy)ex/y. Implicit Differentiation 4. Requires the use of the of . Logarithmic Differentiation 5. Involves finding an extrema for a function.

Optimization Problems 6. Involves setting a derivative equal to zero. 7. Involves the of derivatives. l’Hospital’s Rule 8. Is used to figure out indeterminant forms.

Finding Points of Inflection 9. Is used to figure out indeterminant forms, but can only be used on indeterminant quotients. Others indeterminant forms need some before you can use it.

1. Can be worked out using the quotient rule of derivatives and a trig . Do it!

d 2. Can be worked out from a derivative rule. Do it! tan x = dx 3. Can be worked out using implicit differentiation. Do it!

Z x ln t 4. Can be worked out using a substitution. Do it! dt = 1 t 1 5. − x2 d arctan x = 6. ln x dx 1 7. (ln x)2 2 Z x 1 dt = 1 1 t 8. 1 + x2 9. sec2 x

1. Involves holding a variable fixed (treating it as a constant), and then using the regular derivative rules. Partial Differentiation 2. Involves taking the derivative of an implicit function, and re- quires you to remember that one of the variables is a function Implicit Differentiation of the other.

d 3. a . cos(y) = ex−sin y is dx 4. an implicit derivative.   ∂ dy x−sin y dy cos(y) = ex−sin y is 5. equivalent to − sin y = e 1 − cos y . ∂x dx dx 6. equivalent to 0 = −ex−sin y.

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