Section 1
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find the derivative f'(a) = m
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Cards (79)
Section 1
(50 cards)
find the derivative f'(a) = m
find the slope of the tangent line to f(x) at (a,b)
1/b-a integral a to b v(t)dt or s(b)-s(a) / b-a depending if you know v(t) or s(t)
find the average velocity of a particle on (a,b)
Solve f'(x)= 0 or DNE, make a sign chart, find sign change from positive to negative for relative maximums and evaluate those candidates into f(x) - find lim x--> infinity f(x) and lim x--> -infinity - choose the largest.
find the absolute minimum value of f(x)
Mean Value Theorem - Confirm that f is continuous and differentiable on the interval. - find k and j in (a,b) such that m = f(k) - f(j) / k - j - there is some c in (k,j) such that f'(c) = m
show that there exists a c in (a,b) such that f'(c) = m
find the point where a is th ey-value on f(x) - sketch a tangent line and estimate f'(b) at the point - h'(a) = a/f'(b)
given a graph of f(x) and h(x) = f^-1(x), find h'(a)
- understand that the point (a,b) is on h(x) so the point (b,a) is on f(x) - find b where f(b) = a h'(a) = 1/f'(b)
given the equation for f(x) and h(x) = f^-1(x), find h'(a)
1/f'(a) = m and use y-b m(x-a) - sometimes need to find b = f(a)
find equation of the line normal (perpendicular) to f(x) at (a,b)
Express f '(x) as a fraction and solve for numerator and denominator each equal to zero
find critical vales
find lim x--> infinity and lim x--> infinity ????
find horizontal asymptotes of f(x)
determine where f'(x) is positive (above the x-axis)
given a graph of f'(x), find where f(x) is increasing
Find the equation of the tangent line using y - y, = m (x-x,) where m = f'(0) and the point is (0, f(0)) - then plug in 0.1into this line alternative = y = f'(a) (x-a) + f(a)
Approximate the value of /(0.1) by using the tangent line to f at x = 0
Identify where f'(x) is decreasing
Given a graph of f '(*), find where fG) it concave down.
Use max/min techniques to find values at relative max/mins. Also compare f (a)and f (b) (endpoints)
find range of f(x) on (a,b)
find y values for large values of x ex) 9999999999999
find lim(x) x--> infinity, calculator allowed
Write dy/dx as a fraction. - Set the numerator equal to zero NOTE: be careful to confirm that any values are on the curve. Equation of tangent line is y = b. May have to find b.
find horizontal tangent lines to f(x) or a relation to x and y
1) know the product, quotient, chain rules 2) know derivatives of basic functions - Power Rule: polynomials, radicals, rationals - e^x; b^x - lnx; logx - sinx; cosx; tanx - arcsinx, arccosx, arctanx, sin^-1x, etc.
given the equation for f(x), find its derivative algebraically
show that lim x--> a+f(x) = lim x--> a-; exists and are equal
Show that lim(x) x--> a exists
Express f ''(x) as a fraction and set both numerator and denominator equal to zero. - Make sign chart of f''(x) to find where f''(x) changes sign. (+ to - or - to +) NOTE: be careful to confirm that f (x) exists for any x-values that make f "(x) DNE.
find inflection points algebraically
Identify where f '(x) changes from increasing to decreasing or vice versa.
Given a graph of f '(x), find where /(x) has point(s) of inflection.
Intermediate Value Theorem (IVT) - confirm that f(x) is continuous on (a,b) - show that f(a)<n<f(b)
show that there exists a c in (a,b) such that f(c) = n
f'(a) = m and use y-b m(x-a) - sometimes need to find b = f(a)
find equation of the line tangent to f(x) at (a,b)
Assume domain is (-infinity,+ infinity) Restrictable domains: denominators do not equal 0, square roots of only non-negative numbers, log or ln of only positive numbers, real-world constraints
find the domain of f(x)
Two relationships are required: same slope and point of intersection. - Check that m = f'(x) and and that (x,, y, ) is on both f (x) and the tangent line.
Show that the line y = mx + å is tangent to f (x) at (x,, y,)
Show that f(-x) = -f(x) or f(x) = -f(-x) - symmetric around the origin
show that f(x) is odd
Solve f"(x)= 0 or DNE, make a sign chaft, find sign change from negative to positive for relative minimums and evaluate those candidates along with endpoints back info f '(x) and choose the smallest. - NOTE: be careful to confirm that f (x) exists for any x-values that make f "(x) DNE.
find the minimum slope of a function on (a,b)
Implicit Differentiation - find the derivative of each term - every derivative of y is multiplied by dy/dx - group all dy/dx terms on one side - factor out dy/dx and solve
given a relation of x and y, find dy/dx algebraically
find v(k) and a(k) - signs match = speeding up + - different signs = slowing down -
given v(t), determine if a particle is speeding up at t = k
show that 1) lim x-->a exists (limx-->a- = limx-->a+) Ð f (a) exists 3) limf(x) = f(a)
show that f(x) is continuous
Solve f'(x) = 0 or DNE, make a sign chart, find sign change from negative to positive for relative minimums and evaluate those candidates along with endpoints back into f(x) and choose the smallest. - NOTE: be careful to confirm that f(x) exists for any x-values that make f '(x) DNE.
find the minimum value of a function on (a,b)
Use max/min techniques to find values at relative max/mins. Also compare lim x--> + and - infinity f(x)
find range of f(x) on (-infinity, infinity)
Write dy/dx fraction. Set the denominator equal to zero. NOTE: be careful to confirm that any values are on the curve. Equation of tangent line is x = a. May have to fìnd a.
find vertical tangent lines to f(x) or a relation to x and y
find f'(a)
find instantaneous rate of change of f(x) at a
chain rule f'(g(x)) x g'(x)
find the derivative of f(g(x))
Substitute x = a 1) limit is value if a/c, incl. 0/c = 0, c does not equal 0 2) DNE for a/0 3) 0/0 more work! a) rationalize radicals b) simplify complex fractions c) factor/reduce d) known trig limits 1. lim x --> 0 sinx/x = 1 2. lin x --> 0 1-cosx/x = 0 e) piece-wise function: check if RH = LH at break
find lim x--> a, no calculator
- find the derivative of f'(x) = f''(x) - set numerator and denominator = 0 to find critical points - make sign chart of f''(x) - determine where f''(x) is positive
find interval where the slope of f(x) is increasing
First, be sure that the function is continuous x =a by evaluating each function at x = a. - Then take the derivative of each piece and show that lim x-->a-f'(x) = lim x-->a+f'(x)
show that a piecewise function is differentiable at the point a where the function rule splits
Write the volume formula. Find dV/dt.
Find rates of change for volume problems.
2nd derivative test - find where f'(x) = 0 or DNE - check the value of f''(x) there - if f''(x) = + then it has a minimum - if f''(x) = - then it has a maximum
Find the locations of relative extrema of f(x) given both f''(x) and f'(x) Particularly useful for relations of x and y where finding a change in sign would be difficult.
find f(b) - f(a)/b-a
find average rate of change of f(x) on (a,b)
Find roots. Set function = 0, factor or use quadratic equation if quadratic. graph to find zeros on calculator
find the zeros
Find where lim x--> a+-f(x) = + or - infinity 1) Factor/reduce f(x) and set denominator = 0 2) ln x has VA at x = 0
find verticle asymptotes of f(x)
- find f'(x) - set numerator and denominator = 0 to find critical points - make sign chart of f'(x) - determine where f'(x) is positive
find interval where f(x) is increasing
ratios for rate of change: 1) fast/slow = DNE L >D ?? 2) slow/fast = 0 D>L ?? 3) same/same = Coefficient/coefficient
??? find lim(x) x--> infinity, no calculator
f'(x) = limh-->0 f(x+h) - f(x) / h or f(x) - f(a) / x-a
find f'(x) by the limit definition
Straddle c, using a value, k, greater than c and a value, h, less than c. so f '(c) = f(k) - f(h) / k - h
Given a table of x and f (x) on selected values between a and b, estimate f'(c) where c is between a and b
Identify where f'(x) = 0 crosses the x-axis from above to below OR where f'(x) is discontinuous and jumps from above to below the x-axis
Given a graph of f '(x), find where /(x) has a relative maximum.
Show that f(-x)= f (x) symmetric to y-axis
show that f(x) is even
v(t) = s'(t)
given s(t) (position function), find v(t)
Use TABLE [ASK], find y values for x-values close to a from left and right
Show that lim(x) x--> a exists, calculator allowed
Rolle's Theorem - Confirm that f is continuous and differentiable on the interval. - Find k and j in (a,b) such that f (k) = f (j) then there is some c in (k f'(c) = 0
show that there exists a c in (a,b) such that f'(c) = 0
Section 2
(29 cards)