Calculus I (MAT 145) Dr. Day Friday February 1, 2019 Return Quiz #2/3 and comments Determining Exact Values for Limit Statements Conjugates Common Denominators Expand⇔ Factor ⇔ Reduce Vertical Asymptotes, Horizontal Asymptotes, and Limits What is a Continuous Function? Discontinuities: Removable, Jump, Infinite Date Change for Test #1: Friday, Feb 8: STV 308 Friday, February 1, 2019 MAT 145

the function exists at a. When limit exists If and , then If the left-hand limit exists and the right-hand limit exists and they are the same value, then the overall (two-sided) limit exists. Limits can exist whether or not the function exists at a. And, if the function does exist, but is not the same value as the limit at a (i.e., f(a)≠L), that’s okay, and, in fact, that’s how we know that a function is discontinuous at x=a (stay tuned: Section 2.6). Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Vertical Asymptotes Friday, February 1, 2019 MAT 145

Limits and Vertical Asymptotes Use limit statements to describe the vertical asymptotes. Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

of the curve y = f(x) if either: The line y = L is called a horizontal asymptote of the curve y = f(x) if either: Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Algebra & Limits Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

by finding its intercepts and its limits as and as . Sketch the graph of by finding its intercepts and its limits as and as . Friday, February 1, 2019 MAT 145

This implies three requirements: A function f is continuous at x = a when the following limit statement can be verified at x = a: This implies three requirements: Friday, February 1, 2019 MAT 145

Removable Discontinuity Removable Discontinuity Infinite Discontinuity Jump Discontinuity Friday, February 1, 2019 MAT 145

More about Continuity Friday, February 1, 2019 MAT 145

Identify intervals of continuity & Locations where function is continuous from right or left → → Friday, February 1, 2019 MAT 145

Types of Discontinuities Draw the graph of a function y = f(x) so that the graph shows: exactly one removable discontinuity, and exactly one jump discontinuity, and exactly one infinite discontinuity. After you have completed the graph, identify the location of each of these discontinuities. Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Secant Slope We want to find the tangent line to y = f (x) at the point P (a, f (a)). Consider a nearby point Q (x, f (x)), where x  a, and compute the slope of the secant line PQ: Friday, February 1, 2019 MAT 145

Tangent Slope Then we let Q approach P along the function y = f (x) by letting x approach a. Secant slope is mPQ. Tangent line slope is m. As Q approaches P, x approaches a. And as x approaches a, mPQ approaches m. Friday, February 1, 2019 MAT 145

Tangent slope as limit Friday, February 1, 2019 MAT 145

Another way to label points and calculate slope Let x = a + h and so the slope of the secant line PQ is Friday, February 1, 2019 MAT 145

Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1). Friday, February 1, 2019 MAT 145

C B A D E Friday, February 1, 2019 MAT 145

Calculate the slope at x = −2. Calculate the slope at x = 0. Calculate the slope at x = a. Friday, February 1, 2019 MAT 145

Calculate the slope of f(x) = x2 at x = a. Friday, February 1, 2019 MAT 145

We call this slope calculation the derivative of f at x = a. Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Monday, February 1, 2016 Friday, February 1, 2019 MAT 145 MAT 145

Friday, February 1, 2019 MAT 145

The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of f at x is the limit of the value of the slope function as the secant lines get closer and closer to being a tangent line. Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

The value f ’(a) is called: the derivative of f at x = a, the instantaneous rate of change of f at x = a, the slope of f at x = a, and the slope of the tangent line to f at x = a. Friday, February 1, 2019 MAT 145

The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100? Friday, February 1, 2019 MAT 145

The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100? From the description of the context, the “rate units” are: feet per minute. The value 12 is an input variable, so we are looking at the precise instant that 12 minutes of travel has occurred, since some designated starting time when t = 0. S’ indicates rate of change of S, indicating we have information about how S is changing with respect to t, in feet per minute. The value 100 specifies the rate: 100 feet per minute. Putting it all together: At precisely 12 minutes into the trip, the object’s position is increasing at the rate of 100 feet per minute. Friday, February 1, 2019 MAT 145

The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5? Friday, February 1, 2019 MAT 145

The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5? From the description of the context, the “rate units” are: thousands of dollars per patient. The value 90 is an input variable, so we are looking at the precise instant when 90 patients are in the hospital. C’ indicates rate of change of C, indicating we have information about how C is changing with respect to p, in thousands of dollars per patient. The value 4.5 specifies the rate: 4.5 thousand dollars ($4500) per patient. Putting it all together: At precisely the instant that 90 patients are in the hospital, the cost per patient is increasing at the rate of $4500 per patient. Friday, February 1, 2019 MAT 145

The derivative in action! V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π? From the description of the context, the “rate units” are: cubic cm of volume per cm of radius. The value 3 is an input variable, so we are looking at the precise instant when the sphere’s radius is 3 cm long. V’ indicates rate of change of V, indicating we have information about how V is changing with respect to r, in cubic cm per cm. The value 36π specifies the rate: 36π cubic cm of volume per 1 cm of radius length. Putting it all together: At precisely the instant that the sphere has a radius length of 3 cm, the sphere’s volume is increasing at the rate of 36π cubic cm per cm of radius length. Friday, February 1, 2019 MAT 145

Tangent slope again Friday, February 1, 2019 MAT 145

Find an equation of the tangent line to the parabola y = x2 at the point P(1, 1). Friday, February 1, 2019 MAT 145

Can we create a derivative function f that will be true for any x value where a derivative exists? Friday, February 1, 2019 MAT 145

Calculate the derivative function, f ’(x), for f(x) = x2 Calculate the derivative function, f ’(x), for f(x) = x2. Use the limit definition of the derivative. Friday, February 1, 2019 MAT 145

Friday, February 1, 2019 MAT 145

Here is a graph of the function y = g(x) Here is a graph of the function y = g(x). Arrange the following values in increasing order. Explain your process and determination. Friday, February 1, 2019 MAT 145

Here is the graph of the function y = |x|. Why does the derivative NOT exist at x = 0? Friday, February 1, 2019 MAT 145

Three situations for which a derivative DOES NOT EXIST! Friday, February 1, 2019 MAT 145

For each graphed function, state points at which the function is NOT differentiable. Explain your choices! Friday, February 1, 2019 MAT 145

Match each function, a-d, with its derivative, I-IV. Friday, February 1, 2019 MAT 145

Identify each curve. Explain your choices. Here are the graphs of four functions. One repre- sents the position of a car as it travels, another represents the velocity of that car, a third repre- sents the acceleration of the car, and a fourth graph represents the jerk for that car. Identify each curve. Explain your choices. Friday, February 1, 2019 MAT 145

Here is the graph of a function f. Use it to sketch the graph of f ’. Friday, February 1, 2019 MAT 145