limit & derivatives

Question 1

what is the derivative of the chain rule?

Answer 1

f(x)= u(v(x))
f'(x)= u'(v(x)) v'(x)

Question 2

what is the derivative of an exponential function?

Answer 2

f(x)= e^x
f’(x)= e^x
chain: f(x)= e^u(x)
f’(x)= e^u(x) u’(x)

Question 3

what is the derivative of a logarithmic function?

Answer 3

f(x)= lnx
f'(x)= 1/x
chain: f(x)= lnx
f'(x)= u'(x)/u(x)
*y= bu^(x)
y'=bu^(x) u'(x) lnb

Question 4

what is the derivative of a trigonometric function?

Answer 4

f(x)= sinx                            f(x)= cosx
f'(x)= cosx                          f'(x)=-sinx
f(x)= tanx                            f(x)= cotx    
f'(x)= sec^2 x                     f'(x)=-cosec^2 x
f(x)= secx                           f(x)= cosecx
f'(x)= secxtanx                   f'(x)= -cosecxcotx

Question 5

what is the derivative of a chain trigonometric function?

Answer 5

f(x)= sin u(x)                           f(x)= cos u(x)
f'(x)= cos u(x)  u'(x)                f'(x)=-sin u(x) u'(x)
f(x)= tan u(x)                            f(x)= cot u(x)    
f'(x)= sec^2 u(x) u'(x)              f'(x)=-cosec^2 u(x) u'(x)
f(x)= sec u(x)                           f(x)= cosec u(x)
f'(x)= sec u(x) tan u(x) u'(x)     f'(x)= -cosec u(x) cot u(x) u'(x)

Question 6

what is the derivative of an inverse trigonometric function?

Answer 6

f(x)= arcsinx                     f(x)=arccosx
f'(x)= 1/√(1-x^2)                f'(x)= -1/√(1-x^2)
f(x)= arctanx                    f(x)= arccotx
f'(x)= 1/(1+x^2)                  f'(x)= -1/(1+x^2)

Question 7

what is the derivative of a chain inverse trigonometric function?

Answer 7

f(u(x))= arcsin u(x)                     f(u(x))=arccos u(x)
f'(x)= u'(x)/√(1-u(x)^2)                f'(x)= -u'(x)/√(1-u(x)^2)
f(u(x))= arctan u(x)                    f(u(x))= arccot u(x)
f'(x)= u'(x)/(1+u(x)^2)                  f'(x)= -u'(x)/(1+u(x)^2)

Question 8

what are the steps to find the slope and equation of the tangent line?

Answer 8

Given f(x) & P(Xd, Yd)

1. Calculate f’(x)
2. Calculate m= f’(Xd)
3. Use Xd, Yd & m in y=mx+b to find b
4. Use m & b to write y=mx+b

Question 9

what happens if the tangent line is horizontal?

Answer 9

the slope = zero

Question 10

What is the procedure for finding the intervals where a function is going up or down?

Answer 10

1. Determine f’(x)
2. Set f’(x)=0 and solve for x
3. Construct a table
4. make a statement

Question 11

What is the first derivative test?

Answer 11

1. Determine f’(x)
2. Set f’(x)=0 and solve for x (critical numbers)
3. Construct a table (indicate what intervals are + or - and what is increasing or decreasing)
4. consider the critical number c: if f(x) is decreasing on the left of c and increasing on the right of c then it has a local min. if f(x) is increasing on the left of c and decreasing on the right of c then it has a local max.

Question 12

what is the procedure to determine an inflection point?

Answer 12

1. Determine f’(x)
2. Determine f”(x)
3. Set f”(x)=0 and solve for x
4. Construct a table of x, f”(x) and f(x) (indicate what intervals are + or - and what is concave up and down)
5. Calculate the value of f(x) for the x at which concavity has changes to obtain the inflection points

Question 13

what is the procedure for curve-sketching?

Answer 13

1. (if x=0 in the domain) Calculate the y-intercept by setting x=0 in the equation y= f(x)
2. Calculate the x-intercept by solving f(x)= 0 (if possible)
3. Vertical asymptote (rational functions)
4. Horizontal asymptote (rational functions)
5. Determine f’(x)
6. Set f’(x)=0 and solve for x (critical numbers)
7. Determine at what x values f’(x) is undefined (setting the denominator to zero)
8. Determine f”(x)
9. Set f”(x)=0 and solve for x
10. Determine at what x values f”(x) is undefined (setting the denominator to zero)
11. Construct a table: indicate local max, min, inflection point, increasing, decreasing, concave up and down
12. Use the table to sketch the graph

Question 14

What is the scalar multiple? (limit property)

Answer 14

lim [bf(x)] = b lim f(x) = bL

Question 15

What is the sum or difference? (limit property)

Answer 15

lim [f(x) +/- g(x)] = lim f(x) +/- lim g(x) = L +/- K

Question 16

What is the product? (limit property)

Answer 16

lim[f(x) g(x)] = lim f(x) lim g(x) = LK

Question 17

What is the quotient? (limit property)

Answer 17

lim f(x)/g(x) = limf(x)/limg(x) = L/K (K can’t equal to 0)

Question 18

What is the positive integer exponent? (limit property)

Answer 18

lim [f(x)]^n = [lim f(x)]^n = L^n

Question 19

What is the positive integer root? (limit property)

Answer 19

lim ^n√f(x) = ^n√lim(x) = ^n√L

Question 20

What makes a function continuous? at the point x=c

Answer 20

1. f(c) is defined
2. lim f(x) exists
3. lim f(x)= lim f(c)

Question 21

what is the procedure for logarithmic differentiation?

Answer 21

1. take natural logarithm from both sides
2. simplify using log rules
3. differentiate both sides
4. solve for dy/dx
5. if possible, replace y by its expression in x

Question 22

what is the procedure to determine the maxima or minima of f(x) over [a,b]?

Answer 22

1. Determine f’(x)
2. Set f’(x)=0 and solve for x (critical numbers)
3. Calculate f(x) at all critical values that are in the interval [a,b] as well as the end points of the interval
4. The largest and smallest value from step 3 are the absolute maxima and minima

Question 23

what is the average cost function?

Answer 23

|C (x) = C(x)/x

Question 24

what is the marginal average cost function?

Answer 24

|C’(x)

Question 25

what is the marginal cost/revenue function?

Answer 25

C'(x) and R'(x)

Question 26

what is the revenue function?

Answer 26

R(x)= px, p=price per item

Question 27

what is the profit function?

Answer 27

P(x) = R(x) - C(x)

Question 28

what is the marginal profit function?

Answer 28

P'(x) = R'(x) - C'(x)

Question 29

what is the demand function?

Answer 29

x= f(p)

Question 30

What is the elasticity of demand?

Answer 30

E(p)= -p f'(p) / f(p)

Question 31

What is the second derivative test?

Answer 31

``` Given y=f(x), find maxima and minima 1.Determine f'(x) 2.Set f'(x)=0 and solve for x (critical numbers) say c is a critical number 3. Determine f"(x) 4. Evaluate f"(c) 5. Decision: if f"(c) is bigger than 0, minima if f"(c) is smaller than 0, maxima if f"(c) = 0, test fails ```

Question 32

what is the limit definition?

Answer 32

lim f(x+h) - f(x) / h