Section 4.4 – The Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus If f is continuous on the interval [ a,b ] and F is any function that satisfies F '(x) = f(x) throughout this interval then Alternative forms:

Example 1 Evaluate First Find the indefinte integral F(x): Now apply the FTC to find the definite integral: Notice that it is not necessary to include the “C” with definite integrals

Example 2 Evaluate First Find the indefinte integral F(x): Now apply the FTC to find the definite integral: Notice that it is not necessary to include the “C” with definite integrals

More Examples: New Notation 4. Evaluate F(x) Bounds 5. Evaluate 6. Evaluate If needed, rewrite.

Example 7 Calculate the total area between the curve y = 1 – x 2 and the x -axis over the interval [0,2]. The question considers all area to be positive (not signed area), thus use the absolute value function: Use a integral and the piece-wise function to find the area: Rewrite the equation as a piece-wise function.

Example 8 Assume F '(x) = f (x), f (x) = sin (x 2 ), and F(2) = -5. Find F (1). Use the First Fundamental Theorem of Calculus: We do not have the ability to analytically calculate this integral. It will either be given or you can use a calculator to evaluate the integral.

Example 9 The graph below is of the function f '(x). If f (4) = 3, find f (12). Use the First Fundamental Theorem of Calculus:

White Board Challenge If, for all x, f '(x) = (x – 2) 4 (x – 1) 3, it follows that the function f has: a)a relative minimum at x = 1. b)a relative maximum at x = 1. c)both a relative minimum at x = 1 and a relative maximum at x = 2. d)neither a relative maximum nor a relative minimum. e)relative minima at x = 1 and at x = 2. Multiple Choice

Let F be a function that satisfies the following hypotheses: 1. F is continuous on the closed interval [a,b] 2. F is differentiable on the open interval (a,b) Then there is a number c in (a,b) such that: Mean Value Theorem Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval [a,b] 2. f is differentiable on the open interval (a,b) Then there is a number c in (a,b) such that: Redefine the Conditions Rewrite with integral notation. Solve for the integral.

Mean Value Theorem for Integrals If f(x) is continuous on [a,b], then there exists a value c on the interval [a,b] such that:

Average Value of a Function The average value of an integrable function f(x) on [a,b] is the quantity: This is also referred to as the Mean Value and can be described as the average height of a graph.

Reminder: Average Rate of Change For a ≠ b, the average rate of change of f over time [a,b] is the ratio: Approximates the derivative of a function.

Example 1 Find the average value of f (x) = sin x on [0, π ]. Use the Formula:

Example 2 The height of a jump of a bushbaby is modeled by h (t) = v 0 t – ½gt 2. If g = 980 cm/s 2 and the initial velocity is v 0 =600 cm/s, find the average speed during the jump. Use the Average Value Formula: We are trying to find the average value of SPEED (absolute value of velocity). So we need to find the velocity function. Velocity is the derivative of Position. Thus, the speed function is: Now find when the jump begins and ends (a and b). Evaluate the integral:

Example 2 (Continued) Rewrite the equation as a piece-wise function: Use a integral and the piece-wise function to find the average value: The height of a jump of a bushbaby is modeled by h (t) = v 0 t – ½gt 2. If g = 980 cm/s 2 and the initial velocity is v 0 =600 cm/s, find the average speed during the jump.

White Board Challenge Evaluate

Net Change of a Quantity over a Specified Interval Consider the following problems: 1.Water flows into an empty bucket at a rate of 1.5 liters/second. How much water is in the bucket after 4 seconds? 2.Suppose the flow rate varies with time and can be represented as r(t). How much water is in the bucket after 4 seconds? The quantity of water is equal to the area under the curve of r(t)

Net Change of a Quantity over a Specified Interval Water flows into an empty bucket at a rate of 1.5 liters/second. Suppose the flow rate varies with time and can be represented as r(t). How much water is in the bucket after 4 seconds? The quantity of water is equal to the area under the curve of r(t). Let s(t) be the amount of water in the bucket at time t. Use the First Fundamental Theorem of Calculus: Signed Area under the graph Water in the bucket at 4 s IMPORTANT: If the bucket did not start empty, the integral would represent the net change of water.

Net Change as the Integral of a Rate The net change in s(t) over an interval [t 1,t 2 ] is given by the integral: Integral of the rate of change Net change from t 1 to t 2 Rate at which s(t) is changing Amount of the quantity at t 1

Example 1 If b(t) is the rate of growth of the number of bacteria in a dish measured in number of bacteria per hour, what does the following integral represent? Be specific. The increase in the number of bacteria from hour a to hour c.

Example 2 The number of cars per hour passing an observation point along a highway is called the traffic flow rate q(t) (in cars per hour). The flow rate is recorded in the table below. Estimate the number of cars using the highway during this 2-hour period. t7:007:157:307:458:008:158:308:459:00 q(t)q(t) Since there is no function, we can not use the First Fundamental Theorem of Calculus. Instead approximate the area under the curve with any Riemann Sum (I will use right-endpoints with 0.25 hour lengths):

Example 3 A particle has velocity v(t) = t 3 – 10t t. Without evaluating, write an integral that represents the following quantities: a)Displacement over [0,6] b)Total distance traveled over [3,5]

The Integral of Velocity Assume an object is in linear motion s(t) with velocity v(t). Since v(t) = s'(t) :

White Board Challenge A factory produces bicycles at a rate of: bicycles per week. How many bicycles were produced from the beginning of week 2 to the end of week 3? Week 1Week 2Week 3Week

axb Signed Area = g(x) The Definite Integral as a Function of x Let f be a continuous function on [ a, b ] and x varies between a and b. If x varies, the following is a function of x denoted by g(x) : Notice, g(x) satisfies the initial condition g(a) = 0. Notice that a is a real number.

Example 1 Use the function F(x) to answer the questions below: a)Find a formula for the function. b)Evaluate F(4). c)Find the derivative of F(x). Notice that this is the same as the integral when t = x.

Example 2 Evaluate: In the previous example, in order to find the derivative we had to find the integral: Unfortunately, like many integrals, we can not find an antiderivative for this function. It should be clear there is an inverse relationship between the derivative and the integral. Thus, the derivative of the integral function is simply the original function.

The Second Fundamental Theorem of Calculus Assume that f(x) is continuous on an open interval I containing a. Then the area function: is an antiderivative of f(x) on I ; that is, A'(x) = f(x). Equivalently,

Example 2 (Continued) Evaluate: Since f(x)f(x)

Example 3 Evaluate: Notice the upper limit of the integral is a function of x rather than x itself. We can not apply the 2 nd FTC. But we can find an antiderivative of the integral: Find the derivative of the result:

Example 4 Evaluate: Notice we can not find an antiderivative of the integral AND the upper limit of the integral is a function of x rather than x itself. How do we handle this? Can we apply the 2 nd FTC?

The Upper Limit of the Integral is a Function of x Use the First Fundamental Theorem of Calculus to evaluate the integral: Find the derivative of the result: Chain Rule Constant

Composite Functions and The Second Fundamental Theorem of Calculus When the upper limit of the integral is a function of x rather than x itself: We can use the Second Fundamental Theorem of Calculus together with the Chain Rule to differentiate the integral:

Example 4 (continued) Evaluate: Since,, and f(g(x)) g'(x)

White Board Challenge Find the derivative of the function:

White Board Challenge If h(t) is the rate of change of the height of a conical pile of sand in feet/hour, what does the following integral represent? Be specific. The change in height of the pile of sand from hour 2 to hour 6.

2006 AB Free Response 4 Form B

1991 AB Free Response 1 Let f be the function that is defined for all real numbers x and that has the following properties. i.f ''(x) = 24x – 18 ii.f '(1) = –6 iii.f (2) = 0 a)Find each x such that the line tangent to the graph of f at (x,f(x)) is horizontal. b)Write the expression for f(x). c)Find the average value of f on the interval 1≤x≤3.

2008 AB Free Response 5 Form B

2012 AB Free Response 1