Calculus 1
Math 114/115
Lecture Notes
1 Derivatives
1.1 The Tangent and Velocity Problem (Section 2.1 in text)
Example:
The position of a car is given by the values in the table.
t (seconds) / 0 / 1 / 2 / 3 / 4 / 5S (feet) / 0 / 10 / 32 / 70 / 119 / 178
a) Find the average velocity for the following time periods.
i) 2 – 5 seconds
ii) 2 – 4 seconds
iii) 2 – 3 seconds
b) Sketch a graph of s as a function of t by plotting points and drawing a smooth curve through them.
c) Draw the secant lines whose slopes correspond to the average velocities found in part a).
d) Draw a tangent line that touches the curve at the point t = 2, and use its slope to estimate the instantaneous velocity of the car when t = 2.
[(]
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Example 2 (review of graphs and functional notation): The graph of the function is given. Use the graph of estimate answers to a) – d) below:
a) For which values of x is f(x) < 0?
b) For which values of x does f(x) have a negative slope?
c) For which value(s) of x does f(x) = 0?
d) For which value(s) of x does f(x) have a slope of 0?
e) Calculate each quantity with a calculator, and use the graph of verify your answer:
i) f(0)
ii) f(5)
f) Find the equation of the line that passes through the points A(0, f(0)) and B(5,f(5)). Plot the line. Mark the rise, f(5) – f(0), and the run, 5 – 0, on the graph.
Example 3: The graph of the function f(x) = x2 + 1 is given:
a) Calculate f(1) and plot the point P(1, f(1)) on each of the graphs in part b)
b) Calculate f(1 + h) and plot the point Q(1+ h,f(1+h) ) for the following values of h. In each case draw the secant line PQ and calculate its slope:
i. h = 1
f(1 + h) = ______
Slope of PQ = ______
ii. h = .5
f(1 + h) = ______
Slope of PQ = ______
iii. h = .2
f(1 + h) = ______
Slope of PQ = ______
iv. h = 0
f(1 + h) = ______
Slope of PQ = ______
c) We would like to know the slope of the tangent line at the point P(1,2). You can see that from part b) that as h get smaller, the secant line gets more and more like the tangent line at P. But if we let h = 0 (as in b iv), what problem is encountered?
1.2 The Limit of a Function (section 2.2 in text)
Example 1 : Let
a) What is the largest natural domain of f(x)?
b) Plot a graph of f(x) (simplify f(x) before graphing)
c) Explain why = 5, although f(3) is undefined. [(]
Example 2: Evaluate for each number a and for each function g(x) given below. Follow the steps given::
1. Evaluate g(a). If g(a) is equal to a real number, then that number is the value of the limit, and you are done.
2. If g(a) evaluates to , then try to simplify g(x) and then evaluate g(a) using the simplified version of g(x). If a real number results, then that number is the value of the limit, and you are done.
a)
b)
c)
d)
f)
1.3 Limits Involving Infinity (Section 2.5 in text)
In this section, we consider two types of limits:
1. infinite limits, where f(x) approaches ¥ or -¥
2. limits at infinity, which involve x approaching ¥ or -¥
Note that sometimes f(x) à ¥ as x à ¥ , combining the two types.
Definition: Let f be a function defined on both sides of a (but not necessarily at a itself). If the values of f(x) can be made as large as we please by taking values of x sufficiently close to a (but not equal to a) then be say:
Example 1: Let . In this example, we will find .
a) Can you evaluate f(1)?
b) What happens to f(x) as x gets close to 1? Fill in the following table:
Xà1- / f(x) / xà1+ / f(x)0 / 2
.5 / 1.5
.8 / 1.2
.9 / 1.1
.99 / 1.01
c) Sketch a graph of f(x).
d) Can you find a value of x that makes f(x) > 1,000,000?
Limits at Infinity
In the previous example, we let x approach a particular number, and found that we could make the values of f(x) become arbitrarily large. Now we will let x become large, and see what happens to the values of f(x).
Definition: Let f be a function defined on some interval ( a , ¥ ). Then
means that the values of f(x) can be made as close to L as we like by taking x sufficiently large.
Example 2: Let . In this example, we will find .
a) What happens to f(x) as x becomes large? Fill in the table below and sketch the graph:
b) What is ?
c) How large must you make x in order to make 0 < f(x) < 10-8? [(]
A Summary of Methods for Calculating Limits
Exercise: Evaluate each of the following limits:
a)
b)
c)
d)
e)
Tangents, Velocities and other Rates of Change (Section 2.6 in text)
Example 1: Let f(x) = x2. We will find the equation of the tangent line to f(x) when x = 3:
a) Find the coordinates of the point P and Q. (The x coordinate of P is 3. The coordinates of Q will be in terms of h).
b) Find the slope of the secant line PQ (in terms of h)
c) Express the slope of the tangent line at P as a limit of the expression from part
d) Evaluate the limit from part c) to find the slope of the tangent line through point P
e) Find the equation of the tangent line through point P and sketch the line.[(]
Example 2: We would like to know the slope of the tangent line to the curve f(x) = x2 + 1 at the point P(1,f(1)).
b) If Q is the point Q(1 + h, f(1 + h) ) mark the value 1 + h on the x-axis and f(1 + h) on the y-axis. (we don’t know or care exactly what h is equal to)
c) Find the slope of the secant line that passes through the points P(1,f(1)) and Q(1 + h, f(1 + h) ). (Express your answer not as a number, but in terms of f and h)
d) We can see that as h gets smaller, the slope of the secant line gets closer and closer to the the slope of the tangent line at P, since Q “slides into” P as h gets smaller. What happens to the slope of the secant line if we let h = 0?
e) Since, as we saw in part c), the expression for slope that we came up with in part a) is undefined when h = 0, instead we will take the limit as h approaches zero:
Evaluate by first simplifying the expression and then replacing h with 0.
f) What is the slope of the tangent line at P? Find the equation of the tangent line and draw it.
Example 3: In this exercise will find a general expression for the exact value of the slope of a tangent line. Let P be the point on the graph at which we want to find the slope of the tangent line.
a) Find the coordinates of P and Q (in terms of a, h and f).
b) Find the slope of the secant line PQ (in terms of a, h and f)
c) Express the slope of the tangent line at P as a limit of the expression from part b)
The Derivative (Section 2.7 in text)
Definition: The derivative of a function f at a number a, denoted f’(a) (read as “f prime of a”), is:
if this limit exists.
The derivative may be interpreted in several ways: as the slope of the tangent line to the graph when x=a, as the instantaneous velocity at time a, or, more generally, as the instantaneous rate of change of f(x) with respect to x when x = a.
Example 1: Let f(x) = x - x2
a) Use the graph to arrange the following numbers in increasing order: f’(0), f’(-3), f’(1), f’(½), f’(4), f’(-4);
b) Find and simplify an expression for f’(a)
b) Use the result from part b) to evaluate f’(0), f’(-3), f’(1), f’(½), f’(4), f’(-4); [(]
Example 2: Sketch the graph of a function for which g(0)=0, g’(0) = 3, g’(1) = 0 and
g’(2) = 1
Example 3
Let g(x) = 1 – x3 (graph shown below). Find g’(0) and use it to find the equation of the tangent line to the curve at the point (.5, g(.5) ). Draw the tangent line on the graph below.
The Derivative as a Function (Section 2.8 in text)
Recall that f’(a) is the derivative of the function f(x) at the point x = a, and is equal to the slope of the tangent line to the graph of f(x) at the point
( a, f(a) ) . We can replace the constant a with the variable x to obtain a new function:
This new function, f’(x), is known as the derivative of f(x).
Example 1:
The graph of f(x) (solid line) and its derivative f’(x) (dotted line) are given below.
a)
Mark the points where f’(x) = 0. What do you notice about the graph of f(x) at those points?
b) Mark the interval on which f’(x) is less than zero. What do you notice about the slope of f(x) on that interval? What about the intervals on which f’(x) is greater than zero?
The second derivative:
If f’(x) is the derivative of the function f(x), then the derivative of f’(x), denoted f’’(x), is known as the second derivative of f(x). The second derivative is the rate of change of the first derivative, that is, it is the rate at which the slope of f(x) is changing.
Example 2:
The graphs of f(x), f’(x), and f’’(x) (the dotted straight line) are given below.
a) Mark the interval on which f’’(x) is positive, and that on which it is negative. What does the sign (positive or negative) of f’’(x) tell us about f(x)?
b) Put an x on the spot where f’’(x) = 0. What happens to f’(x) at that point? What happens to f(x)?
Example 3:
The graph of f(x) = - x2 – x + 6 is given below.
a) Find f’(x) using the definition of derivative and plot f’(x) on the graph above.
b) Find f’’(x) using the definition of derivative. Plot f’’(x) on the graph above.
Math 114
Section 2.10
What Does f’ Say About f?
What does f’ say about f?
¨ If f’(x) > 0 on an interval, then f is increasing on that interval
¨ If f’(x) < 0 on an interval, then f is decreasing on that interval
¨ f(x) has a local maximum at a point if f’(x) = 0 at that point, and if f(x) is increasing to the left of the point and decreasing to the right of the point
¨ f(x) has a local minimum at a point if f’(x) = 0 at that point, and if f(x) is decreasing to the left of the point and increasing to the right of the point.
What does f’’ say about f?
¨ If f’’(x) > 0 on an interval, then f is concave upwards on that interval
¨ If f’’(x) < 0 on an interval, then f is concave downwards on that interval[(]
Example:
a) Sketch a graph of a function for which f’ < 0 and f’’ < 0 on all intervals
b) Sketch a graph of a function for which f’ > 0 and f’’ > 0 on all intervals
c) Sketch a graph of a function for which f’ > 0 and f’’ < 0 on all intervals
d) Sketch a graph of a function for which f’ < 0 and f’’ > 0 on all intervals
Example:
The graphs of f’(x) (solid line) and f’’(x) (dotted line) are shown below. Answer the following questions abut the function f(x):
a) On what intervals if f increasing?
b) On what intervals if f decreasing?
c) On what intervals is the graph of f concave up?
d) At what value(s) of x does f have a local minimum?
e) At what value(s) of x does f have a local maximum?
f) Sketch a possible graph of f
Example:
The graphs of f, f’, f’’ are given. Answer the following questions to help identify which graph is which.
a) At approximately which values of x does function 1 have a tangent with slope of zero (flat spots) ?
b) We know that f’ is always equal to the slope of the tangent to f, at any point x. Which function is equal to zero (has x-intercepts) at the same places as function 1 has flat spots?
c) At approximately which values of x does function 2 have a tangent with slope of zero?
d) Which function is equal to zero (has x-intercepts) at the same places as function 2 has flat spots?
e) Identify each graph as f, f’ or f’’: 1______2______3______