5.1 Extreme Values of Functions
What are we learning?
-Absolute extreme values
-Local extreme values
-How to find them
Why?
-Finding maximum and minimum values of functions is
an important issue in real-world problems.
Absolute (Global) Extreme Values (aka - absolute extrema)
Let f be a function with domain D. Then f(c) is the
(a) absolute maximum value on D if and only if f(x) < f(c) for all x in D.
(b) absolute minimum value on D if and only if f(x) > f(c)for all x in D.
Ex. 1: Find the extrema of the sine and cosine functions on [-π/2 , π/2].
Ex. 2: Find the absolute extrema of the function y = x2 on:
a) (-∞,∞)
b) [0,2]
c) (0,2]
d) (0,2)
Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both a maximum value and a minimum value on the interval.
Local Extreme Values (aka - Relative Extrema)
Let c be an interior point of the domain of the function f. Then f(c) is a
(a) local maximum value at c if and only if f(x) < f(c) for all x in some open interval containing c.
(b) local minimum value at c if and only if f(x) > f(c) for all x in some open interval containing c.
Finding Extreme Values
-Extreme values occur where
Local Extreme Values Theorem
If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f ' exists at
c, then f '(c)=0.
*Extreme values occur only at critical points and endpoints.
Ex. 3: Find the absolute maximum and minimum values of f(x) = x 2/3 on [-2, 3].
Ex. 4: Find the extreme values of 𝑓(𝑥) =
1
√4−𝑥 2
.
*Not every critical point or endpoint will be an extreme.
Ex. 5: Find the extreme values of y = x3.
Ex. 6: Find the extreme values of
2
𝑥≤1
𝑓(𝑥) = {5 − 2𝑥
𝑥+2
𝑥>1
5.2 Mean Value Theorem
What are we learning?
-Mean Value Theorem
-Increasing and decreasing functions
-Rolle’s Theorem
Why?
-The mean value theorem connects average and
instantaneous rate of change.
Michel Rolle was a French mathematician who was around when Isaac Newton and Gottfried Leibniz were developing
calculus. Interestingly, he didn’t trust calculus, as he thought it was inaccurate. Despite his faults, he came up with
Rolle’s Theorem, which is the basis of the Mean Value Theorem. The Mean Value Theorem is one of the most important
theorems we will learn in calculus.
Rolle’s Theorem
If f is continuous and differentiable function on [a,b], and f(a)=0 and f(b)=0, then somewhere between a and b, there is
at least one point c where f’(c)=0.
Draw a picture.
Why does f have to be continuous and differentiable?
Rolle’s Theorem is a special case of the Mean Value Theorem.
Mean Value Theorem
If f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point in its interior (a,b), then
there is at least one point c in (a,b) at which
𝑓 ′ (𝑐) =
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
Ex. 1: When the mean value theorem doesn’t work.
Ex. 2: Show that the function y = x2 satisfies the hypothesis of the Mean Value Theorem on the interval [0,2]. Then find
the solution c.
Ex. 3: Let 𝑓(𝑥) = √1 − 𝑥 2 . Find a tangent to f in the interval (-1,1) that is parallel to the secant between (-1,0) and (1,0).
Increasing and decreasing functions
Let f be a function defined on an interval I and let x1 and x2 be any two points in I.
1. f increases on I if x1 < x2 → f(x1) < f(x2).
2. f decreases on I if x1 < x2 → f(x1) > f(x2).
Let f be continuous on [a,b] and differentiable on (a,b).
1. If f’ > 0 at each point of (a,b), then f increases on [a,b].
2. If f’ < 0 at each point of (a,b), then f decreases on [a,b].
Ex. 4: Where is f(x) = x3 – 4x increasing and where is it decreasing?
Other Consequences
When is f’ = 0?
What can you say about functions that have the same derivative?
Ex. 5: Find the function f(x) whose derivative is sinx and whose graph passes through the point (0,2).
Antiderivative
A function F(x) is an antiderivative of a function f(x) is F’(x) = f(x) for all x in the domain of f. The process of finding an
antiderivative is antidifferentiation.
Ex. 6: Find the velocity and position functions of a body falling freely from a height of 0 meters under each of the
following sets of conditions:
a) The acceleration is 9.8 m/s2 and the body falls from rest.
b) The acceleration is 9.8 m/s2 and the body is propelled downward with an initial velocity of 1 m/s.
5.3 Connecting f’ and f’’ with the Graph of f
What are we learning?
-First derivative test
-Concavity
-Second derivative test
Why?
Examining the first and second derivatives allows you to
analyze and explore functions and their graphs.
First Derivative Test for Local Extrema
The following test applies to a continuous function f(x).
At a critical point c:
1. If f’ changes sign from positive to negative at c (f’ > 0 for x<c and f’ < 0 for x>c), then f has a local maximum value at c.
2. If f’ changes sign from negative to positive at c (f’ < 0 for x<c and f’ > 0 for x>c), then f has a local minimum value at c.
3. If f’ does not change sign at c, then f has no local extreme value at c.
At a left endpoint a:
If f’<0 (f’>0) for x>a, then f has a local maximum (minimum) value at a.
At a right endpoint b:
If f’<0 (f’>0) for x<b, then f has a local minimum (maximum) value at b.
Ex. 1: For the following functions, use the First Derivative Test to find the local extreme values. Identify any absolute
extrema.
a) f(x) = x3 – 12x – 5
b) g(x) = (x2 -3)ex
Concavity
The graph of a differentiable function y = f(x) is
a) concave up on an open interval I if y’ is increasing on I.
b) concave down on an open interval I if y’ is decreasing on I.
Concavity Test
The graph of a twice differentiable function y = f(x) is
a) concave up on any interval where y”>0.
b) concave down on any interval where y”<0.
Ex. 2: Determine the concavity of the given functions.
a) y = x2
b) y = 3 + sinx
Point of inflection
A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.
Ex. 3: Find all points of inflection of the graph of y=𝑒 −𝑥
2
Ex. 4: The graph of the derivative of given below. Determine when the function is increasing, decreasing, concave up,
concave down, any local extrema and points of inflection. Sketch a possible graph of f.
Ex. 5: a particle is moving along the x axis with position function
x(t) = 2t3 – 14t2 + 22t – 5
t≥0.
Find the velocity and acceleration, and describe the motion of the particle.
Second Derivative Test for Local Extrema
1. If f’(c)=0 and f”(c)<0, then f has a local maximum at x=c.
2. If f’(c)=0 and f”(c)>0, then f has a local minimum at x=c.
Ex. 6: Find the extreme values of f(x) = x3 -12x -5.
Ex. 7: Let f’(x) = 4x3 – 12x2
a) Identity where the extrema occur.
b) Find the intervals on which f is increasing and decreasing.
c) Find where the graph of f is concave up and concave down.
d) Sketch a possible graph of f.
5.4 Modeling and Optimization
What are we learning?
-Modeling mathematics and real-world situations with
differentiable functions
Why?
-Optimization is used to apply calculus to real world
situations and solve problems involving maximum and
minimum values.
Ex. 1: Find two numbers whose sum is 20 and whose product is as large as possible.
Steps for optimization problems:
1. Determine what extreme value you are trying to find.
2. Write an equation involving the quantity you are trying to maximize/minimize.
3. Make substitutions as needed so that you have a function of a single variable.
4. Determine the domain for the function. What x values will make sense in the problem?
5. Take the derivative and set equal to zero to find critical points.
6. Use the critical points and endpoints to solve the problem.
Ex. 2: An open-top box is to be made by cutting congruent squares of size length x from the corners of a 20 by 25 inch
sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What
is the resulting maximum value?
Ex. 3: You have been asked to design a one liter oil can shaped like a right circular cylinder. What dimensions will use the
least material?
5.5 Linearization and Differentials
What are we learning?
-Linear approximations
-Differentials
-Estimating change
-Absolute, relative, and percentage change
Why?
-Engineering and science depend on approximations in
most practical applications; it is important to
understand how approximation techniques work.
Linearization
If f is differentiable at x=a, then the equation of the tangent line,
L(x) = f(a) + f’(a)(x – a),
defines the linearization of f at a. The approximation f(x) ≈ L(x) is the standard linear approximation of f at a. The point
x=a is the center of the approximation.
Ex. 1: Find the linearization of 𝑓(𝑥) = √1 + 𝑥 at x =0, and use it to approximate √1.02 without a calculator. Then use a
calculator to determine the accuracy of the approximation.
Ex. 2: Find the linearization of f(x) = cosx at x = π/2 and use it to approximate cos 1.75 without a calculator. Then use a
calculator to determine the accuracy of the approximation.
3
Ex. 3: Use linearizations to approximate (a)√123 and (b)√123.
Differentials
Let y = f(x) be a differentiable function. The differential dx is an independent variable. The differential dy is
dy = f’(x) dx
Ex. 5: Find the differential dy and evaluate dy for the given values of x and dx.
a) y = x5 + 37x, x = 1, dx = 0.01
b) y = sinx, x = π, dx = -0.02
c) x + y =xy, x +2, dx = 0.05
Estimating Change with Differentials
Let f(x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is
df = f’(a) dx
Ex. 6: The radius of a circle increases from a = 10 to 10.1 m. Use dA to estimate the increase in the circle’s area A.
Compare the estimate with the true change ΔA and find the approximate area.
5.6 Related Rates
What are we learning?
-Related rate equations
-Solution strategy
Why?
-Related rate problems are another type of real world
problem that calculus is used to solve.
*Any equation involving two or more variables that are differentiable functions of time t can be used to find an equation
that relates their corresponding rates. In related rates problems, we use equations to model a situation in which values
are changing over time. (i.e. Volume of water in a can while it is being filled or the distance between two cars while they
are driving.)
Ex. 1: Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume of the
cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.
Solving Related Rate Problems:
1. Draw a picture, identifying variables and constants.
2. Write down numerical information (in terms of symbols you have chosen).
3. Write down what you are asked to find (usually a rate).
4. Write an equation that relates the variables.
5. Differentiate with respect to time t.
6. Evaluate using known values.
Ex. 2: A hot air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift-off point. At
the moment the range finder's elevation angle is π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the
balloon rising at that moment?
Ex. 3: A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned
the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to
the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the
cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?
Ex. 4: Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a
base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?