Cram deck Flashcards
What is the derivative of a constant
Zero
what is the derivative of (k x f(x))
(k)f’(x)
what is the derivative of (f(x) + g(x))
f’(x) + g’(x)
what is the derivative of (f(x) x g(x))
f(x)g’(x) + f’(x)g(x)
what is the derivative of f(x)/g(x)
g(x)f’(x)- g’(x)f(x)/ g(x)^2
what is the derivtive of sin(f(x)
cos(f(x)) x f’(x)
what is the derivative of cos(f(x)
-sin(f(x) x f’(x)
what is the derivative of tan(f(x)
sec^2 (f(x) x f’(x)
what is the derivative of ln(f(x)
1/f(x) x f’(x)
what is the derivative of e^f(x)
e^f(x) x f’(x)
what is the derivative of a^f(x)
a^f(x) x lna x f’(x)
what is the derivative of sin^-1 f(x)
f’(x)/ (1-f(x)^2)^.5
what is the derivative of cos^-1 f(x)
-f’(x)/ (1 - f(x)^2)^.5
what is the derivative of tan^-1 f(x)
f’(x)/ (1+ f(x)^2)
what is the derivative of f^-1 (x) at x= f(a)
1/ f’(x) at x=a
what is the L’hopitals rule:
If lim x>a f(x)/g(x) = 0/0 and of lim x>a f’(x)/g’(x) exsists then Iim x>a f(x)/g(x) = f’(x)/g’(x)
the same applies for when x> infinity of for a infinity over infinity form
what is a critical point
any c in the domain of f such that either f’(c) =0 or f’(c) = und. This is called a critical point or critical value of f
what is the equation of the tan line to the curve y=f(x) at x=a
y - f(a) = f’(a)(x-a)
what can the tangent line to the graph be used for
can be used to approximate a function value at points very near the point of tangency. This is known as linear approximation. make sure to use the approximation symbol instead of =
what is the equation of the normal line (perpendicular) to the curve f(x) at x=a
y - f(a) = -1/f’(a) x (x-a)
how do you know if a function is increasing/decreasing
if the derivative is positive it is increasing, if its negative its decreasing
how do you know if the curve y=f(x) has a local(relative) minimum at a point where x=c
if the first derivative changes signs from negative to positive at x=c
how do you know if the curve y=f(x) has a local(relative) maximum at a point where x=c
if the first derivative changes signs from positive to negative at x=c
how do you know if y=f(x) is concave upward
if the second derivative is positive on that interval. if it is negative on the interval than it is concave downward
what is the point of inflection
where the concavity of y=f(x) changes
how do you know if y=f(x) has an absolute minimum at x=c on closed interval (a,b)
if f(c) is less than all other values on that interval. it is a maximum if f(c) id greater than all y values on (a,b)
related rates
if several functions of time t are related through an equation such as Pythagorean theorem then we can obtain a relation involving their (time)rates of change by differentiating with respect to x
approximating areas
you can approximate the value of a definite integral. If f is nonnegative on (a,b) then we interpret the area as the region bounded by y=f(x), the x-axis, and lines x=a and x=b. the value of the integrated is approximated by dividing the area up into n number of strips, approximating the area of each strip with a rectangle or other geometric figures, then summing these approximations.
what is the difference between a left sum, right sum, midpoint sum, and a trapaziodal approximation
on the left sum, you start at the left and don’t include the far right. for the right sum, you start right and don’t include the left endpoint. for middle you take the average values of the two-point of delta x. for these 3 you multiple delta x by f(0) + f(1)…. for trapezoidal approximation you multiply by delta x/2 and all of the f(x) values being summed are doubled except for the first and last ones.
what is the integral of k x f(x)dx
k x (integral of) f(x)dx
what is the integral of f(x) + or - g(x) dx
(integral) f(x)dx + or - (integral) g(x)dx
(integral) u^n du =?
u^(n+1)/(n+1) +c
(integral) 1/n du
ln (u) +c
(integral) cos u du
sin u +c
(integral) sin u du
-cos u +c
(integral) tan u du
ln (sec u) +c
(integral) sec^2 u du
tan u +c
(integral) e^u du
e^u +c
(integral) a^u du
a^n/ lna +c
(integral) 1/ (a^2 - u^2)^.5 du
sin^-1 u/a +c
(integral) 1/(a^2 + u^2)
1/a tan^-1 u/a +c
The first fundamental theorem of calculus
if if is continuous on closed interval (a,b) and f’=f, then
integral) from a to b = F(b) - F(a
the second fundamental theorem of calculus
if f is continuous on (a,b) then the function F(x)= (integral) from a to x f(t) dt has a derivative at every point in (a,b) and F’(x) = d/dx (integral) from a to x of f(t) dt = f(x)
(integral) from a to a f(x) =
0
(integral) from a to b f(x) dx is flipped
- (integral) from b to a f(x) dx
(integral) from a to b f(x) dx is split up
(integral) from a to c f(x) dx + (integral) from c to b f(x) dx
what is the area under the curve if there are negative y values
area above - area below.
the volume of a solid of revolution (consisting of disks is given by
Volume= pi (integral) from a to b r^2 dx or dy
the volume of a solid of revolution (consisting of washers) is given by
Volume = pi (integral) from a to b outside R^2 - inside r^2 dx or dy
