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Calculus
Calculus Review "when you see this"..."think this"
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Gravity
Terms in this set (61)
Find the zeros
Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
Show that f(x) is even
show that f(-x) = f(x)
symmetric to y axis
show that f(x) is odd
show that f(-x) = -f(x) OR f(x)=-f(-x)
symmetric around the origin
show that the limit of f(x) as x approaches a exists
show that the limit of f(x) as x approaches a from the left= limit of f(x) as x approaches a from the right; exists and are equal
find the limit of f(x) as x approaches a, calculator allowed
use TABLE [ASK], find y values for x values close to a from left to right
find the limit of f(x) as x approaches a, no calculator
substitute x=a
1) limit is value if b/c, incl. 0/c=0; c does not equal 0
2) DNE for b/0
3)0/0 do more work!
a)rationalize radicals
b)simplify complex fractions
c)factor/reduce
d)known trig limits
1. limit of (sin(x))/x as x approaches 0 is 1
2. lim of (1-cos(x))/x as x approaches 0 is 0
e)piece-wise fan:check if RH=LH at break
find the limit of f(x) as x approaches infinity, calculator allowed
use TABLE[ASK], find y values for large values of x, i.e. .9999999999
find the limit of f(x) as x approaches infinity, no calculator
ratios of rates of changes
1)fast/slow=DNE
2)slow/fast=0
3)same/same=ratio of coefficients
find horizontal asymptotes of f(x)
find the limit of f(x)as x approaches infinity and the limit of f(x) as x approaches negative infinity
find vertical asymptotes of f(x)
find where the limit of f(x) as x approaches a^2 =+- infinity
1)factor/ reduce f(x) and set denominator =0
2)ln x has VA at x=0
find the domain of f(x)
assume domains (-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
show that f(x) is continuous
show that
1)the limit of f(x) as x approaches a exists
2)f(a) exists
3)the limit of f(x) as x approaches a =f(a)
find the slope of the tangent line to f(x) at x=a
find derivative f'(a)=m
find equation of the line tangent to f(x) at (a,b)
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)
same as above but m=-1/f'(a)
find the average rate of change of f(x) on [a,b]
find f(b)-f(a)/b-a
show that there exists a c in [a,b] such that f(c)=n
intermediate value theorem (IVT)
confirm that f(x) is continuous on [a,b], then show that f(a) is less than or equal to n which is less than or equal to f(b)
find the interval where f(x) is increasing
find f'(x), set both numerator and denominator to zero to find critical points, make sign chart of f'(x) and determine where f'(x) is positive
find interval where the slope of f(x) is increasing
find the derivative of f'(x) a=f''(x), set both numerator and denominator to zero to find critical points, make sign chart of f''(x)and determine where f''(x) is positive
find instantaneous rate of change of f(x) at a
find f'(a)
given s(t) (position function), find v(t)
find v(t)=s'(t)
find f'(x) by the limit definition
*frequently asked backwards
f'(x)= lim as h approaches 0 of (f(x+h)-f(x))/h
or
f'(a)=lim as x approaches a (f(x)-f(a))/x-a
find the average velocity of a particle on [a,b]
1/b-a times the integral from a to b of v(t) dt
or
s(b)-s(a)/b-a depending on if you know v(t) or s(t)
given v(t), determine if a particle is speeding up at t=k
find v(k) and a(k). if signs match, the particle is speeding up; if different signs, then the particle is slowing down
given a graph of f'(x), find where f(x) is increasing
determine where f'(x) is positive (above the x-axis)
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
straddle c, using a value, k, greater than c and a value, h, less than c, so f'(c) is about equal to f(k)-f(h)/k-h
given a graph of f'(x), find where f(x) has a relative maximum
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 f(x) is concave down
identify where f'(x) is decreasing
given a graph of f'(x), find where f(x) has points of inflection
identify where f'(x) changes from increasing to decreasing or vice versa
show that a piecewise function is differentiable at the point a where the function rule splits
first be sure that the function is continuous at x=a by evaluating each function at x=a. then take the derivative of each piece and show that the limits of f'(x) as x approaches a from the right and the left are the same
given a graph of f(x) and h(x)=f^-1(x), find h(a)
find the point where a is the y value on f(x), sketch a tangent line and estimate f'(b) at the point, then h'(a)=1/f'(b)
given the equation for 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). so find b where f(b)=a
h'(a)=1/f'(b)
given the equation for f(x), find its derivative algebraically
1) know product/quotient/chain rules
2) know derivatives of basic functions
a) power rule:polynomials, radicals, rationals
b)e^x, b^x
c)ln x; log (little)b x
d)sin x; cos x; tan x
e)arcsin x; arches x; arctan x; sin^-1 x; etc
given a relation of x and y, find dy/dx algebraically
implicit differentiation
find the derivative of each term, using product/quotient/chain appropriately, especially, chain rule: every derivative of y is multiplied by dy/dx; then group all dy/dx terms on one side; factor out dy/dx and solve
find the derivative of f(g(x))
chain rule f'(g(x))g'(x)
find the minimum value of a function on [a,b]
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 slope of a function on [a,b]
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 critical values
express f'(x) as a fraction and solve for numerator and denominator each equal to zero
find the absolute maximum of f(x)
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) , also find the limit of f(x) as x approaches infinity and negative infinity, then choose the largest.
show that there exists a c in [a,b] such that f'(c)=0
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,j] such that f'(c)= 0
show that there exists a c in [a,b] such that f'(c) = m
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 , then there is some c in [k,j] such that f'(c)=m
find the range of f(x) on [a,b]
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 (infinity, -infinity)
use max/min techniques to find values at relative max/mins. also compare the limit of f(x) as x approaches plus or minus infinity
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
Second derivative Test
find where f'(x) =0 OR DNE then check the value of f''(x) there. If f''(x) is positive, f(x) has a relative minimum. if f''(x) is negative, f(x) has a relative maximum.
find inflection points of f(x) algebraically
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
show that the line y=mx+b is tangent to f(x) at (x1, y1)
two relationships are required: same slope and point of intersection. check that m=f'(x1) and that (x1, y1) is on both f(x) and the tangent line
find any horizontal tangent lines to f(x) or a relation of x and y
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 any vertical tangent lines to f(x) or a relation of x and y
write dy/dx as a 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 find a
approximate the value of f(0.1) by using the tangent line to f at x=0
find the equation of the tangent line to f by using y-y1 = m(x-x1) where m = f'(0) and the point is (0,f(0)). then plug in 0.1 into this line; be sure to use an approximate sign.
alternative linearization formula: y=f'(a)(x-a)+f(a)
find rates of change for volume problems
write the volume formula. find dV/dt. careful about product/ chain rules. watch positive(increasing measure)/negative(decreasing measure) signs for rates
find rates of change for pythagorean theorem problems
x^2+y^2=z^2
2xdx/dt+2ydy/dt=2zdz/dt; can reduce 2s
watch positive(increasing distance)/negative (decreasing distance) signs for rates
find the average value of f(x) on [a,b]
find 1/b-a times the integral from a to b of f(x) dxd
find the average rate of change of f(x) on [a,b]
f(b)-f(a)/b-a
given v(t), find the total distance a particle travels on [a,b]
find the integral from a to b of the absolute value of v(t) dt
given v(t), find the change in position a particle travels on [a,b]
find the integral from a to b of v(t) dt
given v(t) and initial position of a particle, find the position at t=a
find the integral from 0 to a of v(t)dt+s(0)
read carefully: starts at ret at the origin means s(0) = 0 and v(0) = 0
d/dx times the integral from a to x of f(t)dt =
f(x)
d/dx times the integral from a to g(x) of f(t)dt =
f(g(x))g'(x)
find area using left riemann sums
A=base[x0+x1+x2+...xn-1]
note: sketch a number line to visualize
find area using right riemann sums
A=base[x1+x2+x3+...xn]
note: sketch a number line to visualize
find area using midpoint rectangles
typically done with a table of values. be sure to use only values that are given. if you are given six sets of points, you can only do 3 midpoint rectangles
note: sketch a number line to visualize
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Verified questions
QUESTION
A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let $v = v(t)$ be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let $M = M(t)$ be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that $F=M \frac{d v}{d t}-u b$ where the force $F = -Mg$. Thus $M \frac{d v}{d t}-u b=-M g$ Let $M_1$ be the mass of the rocket without fuel, $M_2$ the initial mass of the fuel, and $M_0 = M_1 + M_2$. Then, until the fuel runs out at time $t =− M_2/b$, the mass is $M = M_0 - bt.$ Substitute $M = M_0 - bt$ into Equation and solve the resulting equation for v. Use the initial condition $v(0) = 0$ to evaluate the constant.
QUESTION
Show that the points (-2, 9), (4, 6), (1, 0), and (-5, 3) are the vertices of a square.
QUESTION
Evaluate the line integral through C F.dr, where C is given by the vector function r(t). F(x,y,z)=xi+yj+xyk, r(t)=costi+sintj+tl, 0<=t<=pi
CALCULUS
A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm^3/s, how fast is the water level rising when the water is 5 cm deep?