MAT 265 – Review Session Handouts
1. f(x) = 3x – 2 and g(x) = 3 – 10x [Given]
a. (f ·g)(x) = f(g(x))
= f (3-10x)
= 3(3-10x) – 2
= 9 – 90x – 2
= 7 – 90x
b. (g ·f)(x) = g(f(x))
= g (3x – 2)
= 3 – 10(3x – 2)
= 3 – 30x + 20
= 23 – 30x
2. f(x) = 3x – 2, find f -1(x)
Solution:
y = 3x - 2
x = 3y - 2
x + 2 = 3y
x3 + 23 = y
f -1(x) = (x+2)/3
3. Find tangent line to f(x) = 15 – 2x2 at x = 1
Solution:
f(1) = 13
f ΄(x) = -4x
y = -4(x-1) + 13
y = -4x + 17
4. Find limx→1 2-2xx-1
5. Find limx→-2 3x2 + 5x – 9
6. Find limx→-2 x2+4x-12x2-2x
7. limx→∞ 2x4-x2+8x-5x4+7
8. Given the graph of f(x), determine if f(x) is continuous at x = -2, x = 0 and x =3
http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx
For this question, we need to get the limit at that point and also the function value at that point. If they are equal, function is continuous and if they aren’t equal, the function is not continuous.
Take x = -2
f(-2) = 2 limx→-2 f(x) doesn’t exist
Take x = 0
f(0) = 1 limx→0 fx=1
Take x = -3
f(3) = -1 limx→3 f(x) = 0
Derivatives
The derivative of fx with respect to x is the function f'x and is defined as,
f'x = limh→0 fx+h-f(x)h
1. fx = 2x2 – 16x + 35, find f'x using the above definition
f'x = limh→0 2x+h2-16x+h+35-(2x2-16x+35)h
f'x = limh→0 2x2+4xh+2h2-16x-16h+35-2x2+16x-35h
f'x = limh→0 4xh+2h2-16hh
f'x = limh→0 h(4x+2h-16)h
f'x = limh→0 4x + 2h – 16
= 4x – 16
2. ft = tt+1, find the derivative using the above definition
3. Suppose that the amount of fluid in a container at t minutes is given by V(t) = 2t2 – 16t + 35
a. Is the volume of fluid in the container increasing or decreasing at t = 1?
Solution:
V't = 4t – 16
At t = 1,
V'1 = 4(1) – 16 = -12
The rate of change is negative indicating that the volume is decreasing at t =1
b. Is the volume of fluid in the container increasing or decreasing at t = 5 minutes?
c. Is the volume of fluid in the container ever not changing? If yes, then when?
Hint: Derivative must be zero
Some basic derivative rules are as follows:
[fx±g(x)]' = f'x ± g'x
[c fx]' = c f'x
f'c = 0 where c is a constant
fx= xn, then f'x = nxn-1
Examples:
fx = 2x6 + 7x-6
fx = 3x2x-x2
Product rule
[fxg(x)]' = f'xg(x) + g'(x)fx
Quotient rule
[fxgx]' = gxf'x-fxg'(x) gx2
Examples:
a. fx = 3x2x-x2
b. fx = 3x+92-x
Chain rule
Examples:
a. sin (3x2 + x)
Solution:
f'x = cos (3x2 + x) (6x + 1)
= (6x + 1) cos (3x2 + x)
b. y = sec (1 – 5x)
Higher Order derivatives
fx = 5x3-3x2+10x-5
Now, we should be able to differentiate this function easily.
f'x = 15x2-6x+10
f''x = 30x-6
f'''x = 30
Practice: Find first three derivatives of y = cos (x)
Logarithmic Differentiation
Differentiate y = xx
Solution:
Take logarithm of both sides,
ln y = ln xx
ln y = x ln x
Differentiate both sides using implicit differentiation,
y'y=lnx+x 1x
y'=y (lnx+1)
y'= xx (lnx+1)
4. Determine all critical points for the function fx=6x5+33x44-30x3+100
Solution:
5. Determine all numbers c which satisfy the conclusions of the mean value theorem for the following function fx=x3+2x2-x on [-1,2]
Solution:
6. If we want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10 per sq. feet and material used to build the sides cost $6 per sq. feet. If the box must have a volume of 50 sq. feet. Determine the dimensions that will minimize the cost to build box.
7. Evaluate the following
a. limx→0sinxx
b. limt→05t4-4t2-110-t-9t3
8. Determine linear approximation for fx=3x at x = 8. Use linear approximation to approximate the values of 38.05 and 325
9. Compute ∆y and dy if y=cosx2+1-x as x changes from x = 2 to x = 2.03
Integrals/Anti-derivatives
f(x)dx = Fx+c
Here, fx is called the integrand, x is called the integration variable and c is called constant of integration.
Properties
kf(x)dx = kfxdx
-f(x)dx = -fxdx
fx±g(x)dx = fxdx±g(x)dx
Examples
1. 4x10-2x4+15x2x3dx
2. 2secxtanx+16xdx
3. 38y-1e4y2-ydy
4. sec24t3-tan4t3dt
5. xcosx2+1+xx2+1dx
Definite Integrals
If f(x) is a continuous function on [a, b] and also suppose that F(x) is any anti - derivative for f(x). Then
abfxdx = F (b) – F (a)
Examples:
1. -316x2-5x+2 dx
2. Determine the area of region enclosed by y=x2 and y=x