Cram deck

Question 1

What is the derivative of a constant

Answer 1

Zero

Question 2

what is the derivative of (k x f(x))

Answer 2

(k)f’(x)

Question 3

what is the derivative of (f(x) + g(x))

Answer 3

f’(x) + g’(x)

Question 4

what is the derivative of (f(x) x g(x))

Answer 4

f(x)g’(x) + f’(x)g(x)

Question 5

what is the derivative of f(x)/g(x)

Answer 5

g(x)f’(x)- g’(x)f(x)/ g(x)^2

Question 6

what is the derivtive of sin(f(x)

Answer 6

cos(f(x)) x f’(x)

Question 7

what is the derivative of cos(f(x)

Answer 7

-sin(f(x) x f’(x)

Question 8

what is the derivative of tan(f(x)

Answer 8

sec^2 (f(x) x f’(x)

Question 9

what is the derivative of ln(f(x)

Answer 9

1/f(x) x f’(x)

Question 10

what is the derivative of e^f(x)

Answer 10

e^f(x) x f’(x)

Question 11

what is the derivative of a^f(x)

Answer 11

a^f(x) x lna x f’(x)

Question 12

what is the derivative of sin^-1 f(x)

Answer 12

f’(x)/ (1-f(x)^2)^.5

Question 13

what is the derivative of cos^-1 f(x)

Answer 13

-f’(x)/ (1 - f(x)^2)^.5

Question 14

what is the derivative of tan^-1 f(x)

Answer 14

f’(x)/ (1+ f(x)^2)

Question 15

what is the derivative of f^-1 (x) at x= f(a)

Answer 15

1/ f’(x) at x=a

Question 16

what is the L’hopitals rule:

Answer 16

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

Question 17

what is a critical point

Answer 17

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

Question 18

what is the equation of the tan line to the curve y=f(x) at x=a

Answer 18

y - f(a) = f’(a)(x-a)

Question 19

what can the tangent line to the graph be used for

Answer 19

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 =

Question 20

what is the equation of the normal line (perpendicular) to the curve f(x) at x=a

Answer 20

y - f(a) = -1/f’(a) x (x-a)

Question 21

how do you know if a function is increasing/decreasing

Answer 21

if the derivative is positive it is increasing, if its negative its decreasing

Question 22

how do you know if the curve y=f(x) has a local(relative) minimum at a point where x=c

Answer 22

if the first derivative changes signs from negative to positive at x=c

Question 23

how do you know if the curve y=f(x) has a local(relative) maximum at a point where x=c

Answer 23

if the first derivative changes signs from positive to negative at x=c

Question 24

how do you know if y=f(x) is concave upward

Answer 24

if the second derivative is positive on that interval. if it is negative on the interval than it is concave downward

Question 25

what is the point of inflection

Answer 25

where the concavity of y=f(x) changes

Question 26

how do you know if y=f(x) has an absolute minimum at x=c on closed interval (a,b)

Answer 26

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)

Question 27

related rates

Answer 27

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

Question 28

approximating areas

Answer 28

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.

Question 29

what is the difference between a left sum, right sum, midpoint sum, and a trapaziodal approximation

Answer 29

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.

Question 30

what is the integral of k x f(x)dx

Answer 30

k x (integral of) f(x)dx

Question 31

what is the integral of f(x) + or - g(x) dx

Answer 31

(integral) f(x)dx + or - (integral) g(x)dx

Question 32

(integral) u^n du =?

Answer 32

u^(n+1)/(n+1) +c

Question 33

(integral) 1/n du

Answer 33

ln (u) +c

Question 34

(integral) cos u du

Answer 34

sin u +c

Question 35

(integral) sin u du

Answer 35

-cos u +c

Question 36

(integral) tan u du

Answer 36

ln (sec u) +c

Question 37

(integral) sec^2 u du

Answer 37

tan u +c

Question 38

(integral) e^u du

Answer 38

e^u +c

Question 39

(integral) a^u du

Answer 39

a^n/ lna +c

Question 40

(integral) 1/ (a^2 - u^2)^.5 du

Answer 40

sin^-1 u/a +c

Question 41

(integral) 1/(a^2 + u^2)

Answer 41

1/a tan^-1 u/a +c

Question 42

The first fundamental theorem of calculus

Answer 42

if if is continuous on closed interval (a,b) and f'=f, then | integral) from a to b = F(b) - F(a

Question 43

the second fundamental theorem of calculus

Answer 43

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)

Question 44

(integral) from a to a f(x) =

Answer 44

0

Question 45

(integral) from a to b f(x) dx is flipped

Answer 45

- (integral) from b to a f(x) dx

Question 46

(integral) from a to b f(x) dx is split up

Answer 46

(integral) from a to c f(x) dx + (integral) from c to b f(x) dx

Question 47

what is the area under the curve if there are negative y values

Answer 47

area above - area below.

Question 48

the volume of a solid of revolution (consisting of disks is given by

Answer 48

Volume= pi (integral) from a to b r^2 dx or dy

Question 49

the volume of a solid of revolution (consisting of washers) is given by

Answer 49

Volume = pi (integral) from a to b outside R^2 - inside r^2 dx or dy