sign in sign up
<<
Home , Implicit function, Quotient rule, Related rates

Barry McQuarrie’s I Glossary & Technique Quiz

Instructions: For each group (groups are separated by horizontal lines), match term or quantity in left column to de- scriptions that apply 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 6 x→a 2. A vertical or horizontal line on a graph which a Asymptote 2 approaches. 3. Measures how a function bends, or in other words, the func- Vertical Asymptote 1 tion’s . 4. If lim f(x) = L, then y = L is called this. Concave Up 5 x→±∞ 5. If the function f(x) is above its line at x = a, then Horizontal Asymptote 4 the function has this property at x = a. The function has this property if f 00(a) > 0. Point of Inflection 7 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 3 Concavity possible if f 00(a) = 0. 8. If the function f(x) is below its tangent line at x = a, then Concave Down 8 the function has this property at x = a. The function has this property if f 00(a) < 0. Absolute Maximum 10 9. Any of the maximum or minimum values for a function. Extrema 9 10. The largest value the function takes over its entire domain.

1. This quantity involves a constant, usually something like Antidifferentiation 4 g(x) + C (although other forms are possible), and when you assign different values to the constant C you get different . dy/dx 7 2. The process of finding the derivative of a function f(x). 3 3. A family of curves.

Differentiation 2 4. The process of finding an antiderivative of a function f(x). 5. Informally, this quantity can be written as dx and represents 6 a small amount of x. 6. This quantity is included when an antidifferentiation is per- Differential 5 formed. 7. This quantity represents the instantaneous rate of change 1 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 ex+y = 2 3. (x + y)ex+y−1 4. x − y ln(ex − ey) = 7 5. xy ln(ex−y) = 4 x 6. y 7. ln(ex − ey)

1. This quantity is always non-negative. If v(t) is the velocity, Z t2 Displacement 3 this quantity is given by |v(t)| dt. t1 g(x + h) − g(x) Indeterminant Form 2 2. lim h→0 h

Differentiation 4 3. How far a particle has moved during a time interval from t1 to t2. If v(t) is the velocity, this quantity is given by Z t2 Distance Traveled 1 v(t) dt. t1 4. Mathematically easy. Integration 5 5. Mathematically hard.

x − x0 x1 − x0 of tangent line to a at a 1. = y − y0 y1 − y0 point 3 2. The derivative of the function.

Equation of a straight line 1, 7 3. The derivative of the function evaluated at the point. 4. z = f(x, y) 2 1, 5, 7 in R 5. f(x, y) = 0

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

7. y − y0 = m(x − x0) in R3 4 8. x = f(t), y = g(t), t > 0

Space curve in R3 9 9. x = f(t), y = g(t), z = h(t), t > 0

1. − cos(x3) d sin(x3) = 4 dx 2. cos(x3)

2 3 Z 3. −3x cos(x ) 3 x2 sin(x3) dx = 1 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 2 ated it yields a family of curves as the solution.

Definite 7 2. The process of evaluating an integral (definite, indefinite, or improper) of a function f(x). Indefinite Integral 1 Z b 3. Formally, this quantity looks like f(x) dx where the in- a 3 tegrand f(x) is infinite for some x ∈ [a, b], or a → −∞ or b → ∞. Integral 5 Z b 4. For f(x) dx, this quantity is f(x). a Limits of Integration 6 5. Used as a way to refer to any of the specific types of . Z b Integrand 4 6. For f(x) dx, this quantity is a and b. a d Z x Z b FTC Part 1: w(t) dt = 8 7. Formally, this quantity looks like f(x) dx. When evalu- dx a a ated it yields a number. Z b d 8. w(x) FTC Part 2: [w(x)] dx = 9 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 = 5 dx g(x) g(x)f 0(x) − f(x)g0(x) 6. g(x) d 10 [f(x)g(x)] = g0(x)f 0(x) − f(x)g(x) dx 7. (g(x))2 d 3 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 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 3, 4 4. Requires the use of the of . Logarithmic Differentiation 1, 4 5. Involves finding an extrema for a function. 6. Involves setting a derivative equal to zero. Optimization Problems 5 7. Involves the of derivatives. 9 l’Hospital’s Rule 8. Is used to figure out indeterminant forms. 9. Is used to figure out indeterminant forms, but can only be Finding Points of Inflection 6 used on indeterminant quotients. Others indeterminant forms need some work before you can use it.

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

d 2. Can be worked out from a derivative rule. Do it! tan x = 1, 9 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 = 4, 7 1 t 1 5. − x2 d arctan x = 3, 8 6. ln x dx 1 7. (ln x)2 Z x 2 1 2, 6 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 1 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 2 of the other. 3. a . d cos(y) = ex−sin y is 4, 5 dx 4. an implicit derivative.   dy x−sin y dy ∂  x−sin y 5. equivalent to − sin y = e 1 − cos y . cos(y) = e is 3, 6 dx dx ∂x 6. equivalent to 0 = −ex−sin y.

Page 4 of 4