This document contains notes from a calculus class covering topics including: implicit differentiation, related rates, linear approximations, maximum and minimum values, the mean value theorem, limits at infinity, and curve sketching. Example problems are provided for each topic to demonstrate key concepts and techniques.

1. CALC 224 FINAL REVIEW
12.11.2015
2.6 Implicit Differentiation (EMILY)
ifferentiate all terms and add after each y term that has been differentiatedD dx
dy
Example​ : Differentiate with respect to x.5x2 + y2 = 2
Differentiate the following with respect to x.
1. yx2 + x − y2 = 4
2. cos xsin y4 = 1
3. os xy in yc = 1 + s
Find the equation of the line tangent to at the pointsin 2x cos 2yy = x , )(2
π
4
π
2.7 Rates of Change in the Natural and Social Sciences (TORI)
The height (in meters) of a projectile shot vertically upward from a point 2 meters above ground level
with an initial velocity of 24.5m/s is after ​t​ seconds4.5t .9th = 2 + 2 − 4 2
a. Find the velocity after 2s and after 4s.
b. When does the projectile reach its maximum height?
c. What is the maximum height?
d. When does it hit the ground?
e. With what velocity does it hit the ground?
A spherical balloon is being inflated. Find the rate of increase of the surface area ( ) withπrS = 4 2
respect to the radius ​r​ when ​r​ is
a. 1ft
b. 2ft
c. 3ft
d. What conclusion can you make based on the answers to a-c?

2. 2.8 Related Rates (TORI)
Example: Air is being pumped into a spherical balloon so that its volume increases at a rate of
100cm^3/s. How fast is the radius of the balloon increasing when the diameter is 50cm?
1. A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If
water is being pumped into the tank at a rate of 2m^3/min, find the rate at which the water
level is rising when the water is 3m deep.
2. The length of a rectangle is increasing at a rate of 8cms and its width is increasing at a rate of
3cm/s. When the length is 20cm and the width is 10 cm, how fast is the area of the rectangle
increasing?
3. A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight
toward the building at a speed of 1.6m/s, how fast is the length of his shadow on the building
decreasing when he is 4m from the building?
2.9 Linear Approximations and Differentials (EMILY)
(x) (a) (a)(x ) a is givenL = f + f′
− a
y (x)dxd = f′

3. Example: Find the linearization of the function at a = 1 and use it approximate .(x)f = √x + 3 √3.98
Find the linearization L(x) of the function at a.
1. (x) x , af = x4 + 3 2 = − 1
2. (x) in x , af = s = 6
π
Find the differential and evaluate ​dy for the given values of ​x and ​dx.
1. an x , x , dx .1y = t = 4
π = − 0
2. , x , dx .05y = x−1
x+1 = 2 = 0
3.1 Maximum and Minimum Values (EMILY)
The Extreme Value Theorem: If f is continuous on a closed interval [a,b], then f attains an absolute
maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b].
Fermat’s Theorem: If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c) = 0.
A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) does
not exist.

4. Example: Find the absolute maximum and minimum values of the function
(x) x ≤x≤4f = x3 − 3 2 + 1 − 2
1
Find the critical numbers of the function.
1. (x) x x 6xf = 2 3 − 3 2 − 3
2. (x) cos θ θf = 2 + sin2
Find the absolute maximum and absolute minimum values of f on the given interval.
1. (x) cos t in 2t [0, ]f = 2 + s 2
π
2. (x) [− , ]f = (x )2 − 1
3
1 2
Sketch the following…
a. A function that has a local maximum at 2 and is differentiable at 2
b. A function that has a local maximum at 2 and is continuous but not differentiable at 2
c. A function that has a local maximum at 2 and is not continuous at 2
3.3 The Mean Value Theorem (TORI)
If f’(x) > 0, then f is increasing on that interval, if f’(x) < 0 on an interval, then f is decreasing on that
interval.
If f’ changes from positive to negative at c, then f has a local maximum at c.
If f’ changes from negative to positive at c, then f has a local minimum at c.

5. If f’ does not change sign, then f has no local maximum or minimum at c.
If f’’(x) > 0 for all x in I, then the graph of f is concave upward on I.
If f’’(x) < 0 for all x in I, then the graph of f is concave downward on I.
An inflection point is a point where the concavity of the function changes (from concave up to down
or from concave down to up).
If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at c.
If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at c.
Find the intervals on which f is increasing or decreasing, find the local maximum and minimum values
of f, find the intervals of concavity and the inflection points.
1. (x) x x 6xf = 2 3 + 3 2 − 3
2. (x) xf = x4 − 2 2 + 3
3. (x) in x os x , 0≤x≤2πf = s + c
3.4 Limits at Infinity; Horizontal Asymptotes (EMILY)
he line y is called a horizontal asymptote of the curve f(x)if either lim f(x) or lim f(x)T = L x→∞ = L x→ −∞ = L
Example: Find the limit (or show that it does not exist) of (x )limx→−∞
4 + x5
Find the limit or show that it does not exist.
1. ( xlimx→ ∞ √9x2 + x − 3
2. limx→−∞ x +14
1+x6
3. (x )limx→−∞ + √x x2 + 2

6. 3.5 Curve Sketching (TORI)
Sketch the curve – include all relevant information.
1. (x )y = x − 4 3
2. y = x−x2
2−3x+x2
3. y = x
x −92
4. y = √x2 + x − 2
5. os xy = x + c