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AP Calculus Review Ch3

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Section 1

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instantaneous rate of change

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Calculus Math

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Cards (37)

Section 1

(37 cards)

instantaneous rate of change

Front

of f with respect to x at a is the derivative f'(a)=the limit of (f(a+h)-f(a))/h as h approaches 0

Back

nth derivative

Front

when the number of primes exceeds 3, it is replaced with the list of ^n-x

Back

Intermediate Value Theorem for Derivatives

Front

If a and b are any two points in an interval on which f is differentiable, then f' takes on every value between f'(a) and f'(b)

Back

Power Rule for Arbitrary Real Powers

Front

If u is a positive differentiable function of x and n is any real number, then uⁿ is a differentiable function of x and d(uⁿ)/dx = nuⁿ⁻¹(du/dx)

Back

velocity

Front

the derivative of the position function

Back

inverse function-inverse cofunction identities

Front

arccosx= π/2- arcsinx arccotx= π/2- arctanx arcscsx= π/2- arcsecx

Back

Power Rule for Negative Integer Powers of x

Front

If n is a negative integer and x does not equal 0 ,then d(x^n)/dx =nx^n-1

Back

jerk

Front

the derivative of acceleration or the third derivative of the position function

Back

Power Rule for Positive Integer Powers of x

Front

If n is a positive integer, then d(x^n)/dx = nx^n-1

Back

orthogonal curves

Front

the technical word for "crossing at right angles"

Back

Product Rule

Front

The product of two differentiable functions u and v is differentiable, and d(uv)/dx = u(du/dx)+v(dv/dx)

Back

instantaneous velocity

Front

the derivative of the position function s=f(t) with respect to time

Back

derivative of f at a

Front

the function f' whose value at x is (f(x+h)-f(x))/h as h approaches 0

Back

differentiable function

Front

a function that is differentiable at every point of its domain

Back

Power Chain Rule

Front

d(uⁿ)/dx = nuⁿ⁻¹(du/dx)

Back

simple harmonic motion

Front

ex. the motion of a weight bobbing up and down on the end of a spring

Back

free-fall constants

Front

English: g=32 ft/sec², s=1/2(32)t²=16t² Metric:g-9.8m/sec², s=1/2(9.8)t²=4.9t²

Back

Sum and Difference Rule

Front

If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, d(u+-v)=du/dx +- dv/dx

Back

right-hand derivative

Front

the definition of the derivative as x approaches from the right side (+)

Back

speed

Front

the absolute value of velocity

Back

differentiable on a closed interval

Front

a function that has a derivative at every interior point of its interval and if there are right-hand and left-hand limits

Back

left-handed derivative

Front

the limit of the definition of the derivative as x approaches 0 from the left side (-)

Back

symmetric difference quotient

Front

the same value of h will usually yield a better approximation when the equation (f(a+h)-f(a-h))/2h is used

Back

average velocity

Front

of the object over the time interval is vav= displacement/time= ∆s/∆t= (f(t+∆t)-f(t))/∆t

Back

implicit differentiation

Front

the process of finding dy/dx in terms of x and y together

Back

Constant Multiple Rule

Front

If u is a differentiable function of x and c is a constant, then d(cu)dx = c(du/dx)

Back

normal to the surface

Front

the line perpendicular to the surface at a point of entry

Back

displacement

Front

of the object over the time intervan from t to t+∆t is ∆s=f(t+∆t)-f(t)

Back

Chain Rule

Front

If f is differentiable at the point u=g(x), and g is differentiable at x, then the composite function (f₀g)(x)=f(g(x)) is differentiable at x, and (fog)'(x)=f'(g(x))g(x) In Leibniz notation dy/dx = dy/dudu/dx

Back

numerical derivative (NDER)

Front

a graphing calculator program to solve a derivative that emphasizes both the function and the point

Back

logarithmic differentiation

Front

used to simplify differentiation by taking the natural log of both sides of the equation

Back

Derivative of a Constant Function

Front

If f is the function with the constant value c, then df/dx = d(c)/dx = 0j

Back

local linearity

Front

a function that is differentiable at a closely resembles its own tangent line very close to a

Back

acceleration

Front

when the limit of (f(a+h)-f(a))/h as h approaches 0 exists

Back

marginal cost

Front

dc/dx= the limit of (c(x+1)-c(x))/h as h approaches 0

Back

Quotient Rule

Front

At a point where v does not equal 0, the quotient y=u/v of two differentiable functions is differentiable, and d(u/v)/dx = (v(du/dx)-u(dv/dx))/v^2

Back

sensitivity to change

Front

when a small change in x results in a large change in f(x)

Back

AP Calculus Review Ch3

Preview this deck

instantaneous rate of change

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Cards (37)

instantaneous rate of change

of f with respect to x at a is the derivative f'(a)=the limit of (f(a+h)-f(a))/h as h approaches 0

nth derivative

when the number of primes exceeds 3, it is replaced with the list of ^n-x

Intermediate Value Theorem for Derivatives

If a and b are any two points in an interval on which f is differentiable, then f' takes on every value between f'(a) and f'(b)

Power Rule for Arbitrary Real Powers

If u is a positive differentiable function of x and n is any real number, then uⁿ is a differentiable function of x and d(uⁿ)/dx = nuⁿ⁻¹(du/dx)

velocity

the derivative of the position function

inverse function-inverse cofunction identities

arccosx= π/2- arcsinx arccotx= π/2- arctanx arcscsx= π/2- arcsecx

Power Rule for Negative Integer Powers of x

If n is a negative integer and x does not equal 0 ,then d(x^n)/dx =nx^n-1

jerk

the derivative of acceleration or the third derivative of the position function

Power Rule for Positive Integer Powers of x

If n is a positive integer, then d(x^n)/dx = nx^n-1

orthogonal curves

the technical word for "crossing at right angles"

Product Rule

The product of two differentiable functions u and v is differentiable, and d(uv)/dx = u(du/dx)+v(dv/dx)

instantaneous velocity

the derivative of the position function s=f(t) with respect to time

derivative of f at a

the function f' whose value at x is (f(x+h)-f(x))/h as h approaches 0

differentiable function

a function that is differentiable at every point of its domain

Power Chain Rule

d(uⁿ)/dx = nuⁿ⁻¹(du/dx)

simple harmonic motion

ex. the motion of a weight bobbing up and down on the end of a spring

free-fall constants

English: g=32 ft/sec², s=1/2(32)t²=16t² Metric:g-9.8m/sec², s=1/2(9.8)t²=4.9t²

Sum and Difference Rule

If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points, d(u+-v)=du/dx +- dv/dx

right-hand derivative

the definition of the derivative as x approaches from the right side (+)

speed

the absolute value of velocity

differentiable on a closed interval

a function that has a derivative at every interior point of its interval and if there are right-hand and left-hand limits

left-handed derivative

the limit of the definition of the derivative as x approaches 0 from the left side (-)

symmetric difference quotient

the same value of h will usually yield a better approximation when the equation (f(a+h)-f(a-h))/2h is used

average velocity

of the object over the time interval is vav= displacement/time= ∆s/∆t= (f(t+∆t)-f(t))/∆t

implicit differentiation

the process of finding dy/dx in terms of x and y together

Constant Multiple Rule

If u is a differentiable function of x and c is a constant, then d(cu)dx = c(du/dx)

normal to the surface

the line perpendicular to the surface at a point of entry

displacement

of the object over the time intervan from t to t+∆t is ∆s=f(t+∆t)-f(t)

Chain Rule

If f is differentiable at the point u=g(x), and g is differentiable at x, then the composite function (f₀g)(x)=f(g(x)) is differentiable at x, and (fog)'(x)=f'(g(x))*g(x) In Leibniz notation dy/dx = dy/du*du/dx

numerical derivative (NDER)

a graphing calculator program to solve a derivative that emphasizes both the function and the point

logarithmic differentiation

used to simplify differentiation by taking the natural log of both sides of the equation

Derivative of a Constant Function

If f is the function with the constant value c, then df/dx = d(c)/dx = 0j

local linearity

a function that is differentiable at a closely resembles its own tangent line very close to a

acceleration

when the limit of (f(a+h)-f(a))/h as h approaches 0 exists

marginal cost

dc/dx= the limit of (c(x+1)-c(x))/h as h approaches 0

Quotient Rule

At a point where v does not equal 0, the quotient y=u/v of two differentiable functions is differentiable, and d(u/v)/dx = (v(du/dx)-u(dv/dx))/v^2

sensitivity to change

when a small change in x results in a large change in f(x)

If f is differentiable at the point u=g(x), and g is differentiable at x, then the composite function (f₀g)(x)=f(g(x)) is differentiable at x, and (fog)'(x)=f'(g(x))g(x) In Leibniz notation dy/dx = dy/dudu/dx