AP Calculus AB: Semester Review Notes Information in the box are MASTERY CONCEPTS. Be prepared to apply these concepts on your midterm. Name: Date: Period: I. Limits and Continuity Definition of Average Rate of Change ( Slope) f ( b) f ( a) y1 y The average rate of change of y f (x) from a to b is y Geometrically the b a x1x average rate of change is the slope of the secant line. slope formula If f (t) is the position function of a particle that is moving on a straight line, then the average velocity of the particle over the time interval x < time < c is displacement f ( c) Average velocity = time x c Example 1: The displacement of a particle moving in a straight line is given by the equation of motion f ( t) t t 4. Find the average velocity of the particle over the time interval 0 < t < 5. Definition of One-Sided Limits & The Existence of a Limit The right-hand it means that x approaches c from values greater than c. L x c The left-hand it means that x approaches c from values less than c. L x c The it of f (x) as x approaches c is L if and only if the right and the left hand its are both equal to L. Limits that FAIL to Exists The left and the right hand its are NOT EQUAL. The y-values increase without bound. The y-values oscillate infinitely between y = -1 to y = 1
Definition of Continuity A function f is continuous at c if the following three conditions are met. (1) f (c) exists (1) The y-value exists at x = c () x c exists () The it exists as x approaches c () x c f ( c) () The it equals the y-value Example : (a) f( x) x (b) f( x) x (c) f( x) x (d) f ( ) (f) f( x) x (g) f( x) x (h) f( x) x (i) f () (j) Is the function continuous at x = -? At x =? Example : Find the it. If the it is infinite, then say, The it does not exists. (a) x 5 (b) x ( x x 4) (c) x 0 (ln x) (d) x 4 ( x x 4) ( x 4) (e) x 1 x 8 x 1
Example 4: For what values of a is x f (x) ax 10 x continuous? x Example 5: Given the function ax xt () bx 4 x x If this function is differentiable at x =, then what is the value of a= and b =. What is the sum a + b? II. Differentiation Definition of the derivative of a function The derivative of a function f at x, denoted f (x) is f x x f x x 0 ( ) ( ) x if the it exists. Definition of the derivative at a point If we replace x = a in the definition of the derivative above we get a numerical value which is equal to the derivative at that point x = a. f ( a x) f ( a) f (x a) is x 0 x The derivative is always the it of the slope of the secant line as the denominator goes to zero. y1 y f ( a) An alternate way to write f (x a) = x1 x = x a x x x a 1 The derivative is the slope of the tangent line at x = a. The derivative is also the Instantaneous Rate of Change of f(x) with respect to x when x = a.
Example 6: What is the difference between the AVERAGE Rate of Change and the INSTANTANEOUS Rate of Change? Example 7: Use the definition of the derivative to find f (x). x x How do you approximate the derivative? Use any slope formula to approximate the value of the derivative at x = c. You want to select small values of h so that x+h is CLOSE to x. Select: x = c and h = 0.01 or h = 0.001 Forward difference quotient Backwards Difference Quotient Symmetric Difference Quotient f ( x h) f ( x h) f ( x h) f ( x h) h h h The symmetric difference quotient provides the best approximation of the derivative. Hence, the TI Calculators use the symmetric difference quotient with h=0.001 to find numerical approximations of the derivative. The syntax is nderv( f(x), x, c, h). Differentiability implies Continuity A function is differentiable at x = a only if f (x a) is a REAL NUMBER (that is, it EXISTS) If a function is differentiable at x = a, then the function is also continuous there. If the function is NOT continuous at x = a, then the function is also NOT differentiable there.
There are three possible ways for a function NOT to be differentiable at x = a. CUSP at x = a NOT Continuous at x = a Vertical tangent line at x = a Here the derivative is infinite so we say that it does not exist. Example 8: For what value of x in the interval -6 < x < 6, is the function NOT differentiable? Differentiation Rules: a) uv) (1) The constant rule 0 (4) Product rule uv uv u ) af(x) ) () Constant times a function af ( x) (5) The Quotient rule v uv uv v () The Power rule d n ( x ) n1 nx Example 9: Find the first derivative and then find the second derivative. Box both answers. 1 5 1 (a) x x 4 (b) x 5 15 x x x 5
Example 10: Find the first derivative and simplify. 4 (a) ( x )( x 7x) (b) 4x x x Differentiation Rules Continued: The Chain Rule The Chain Rule is used on composite functions: y f (u) and u g(x) n f ( u)) u ) n1 (6) The Chain Rule f ( u) u (7) The Power Chain Rule nu u Example 11: Find the derivative. (a) x 4x 7 (b) x 5 f( x) x 1 Differentiation Rules Continue: The Six Trigonometric Functions; where u g(x) sin u) (8) cos( u) u tanu) (10) sec ( u) u secu) (1) sec( u) tan( u) u cosu) (9) sin( u) u cot u) (11) csc ( u) u cscu) (1) csc( u)cot( u) u Example 1: Find the derivative and simplify. (a) csc ( x 4 y ) (b) y x tan(x) xcot( x)
The Process: Implicit Differentiation : Use when you CAN NOT solve for y but you need to find y. (1) Take the derivative of both sides of the equation with respect to x () Collect all the y terms on the LEFT side of the equation and move all other terms to the RIGHT side of the equation. () FACTOR out y (4) Divide to solve for y Recall: (1) x 1 and () use the chain rule to get y Example 1: Find the derivative, y (a) 4 sin( xy 4 4 y y ) (b) x y 5xy 7 III. Applications of Derivatives Tangent Lines The equation with slope = m that passes through the point ( x 1, 1 Use: m = f (x 1 ) because the derivative is the slope of the tangent line y ) is ( y y1) m( x x1 ) Example 14: Write the equation of the tangent line to the graph x 4x 7 at the point (, 7)
Example 15: Write the equation of the tangent lines to the curve xy x 1 y that are parallel to the line x 1 Test for Increasing and Decreasing Functions (1) A function f is increasing on an interval if 0 on that interval () A function f is decreasing on an interval if 0 on that interval Test for Concavity (1) If 0 on an interval, then the function is concave up on that interval. () If 0 on an interval, then the function is concave down on that interval. Example 16: For the function (a) Find the first derivative. 5 65 x x 0x for all real numbers. (b) Find the second derivative. (c) For what values of x is the function increasing? Decreasing? (d) For what values of x is the function concave up? Concave down?
Particle Motion along a straight line x(t) the position of the particle v ( t) the velocity of the particle SPEED = velocity dt dv d x a( t) dt dt the acceleration of the particle The particle is moving right (increasing) when the velocity is POSITIVE The particle is moving left (decreasing) when the velocity is NEGATIVE The particle is SPEENDING UP when the velocity & acceleration are the SAME SIGN The particle is SLOWING DOWN when the velocity & acceleration are OPPOSITE SIGNS Example 17: A particle starts moving at time t = 0 and moves along the x-axis so that its position at time t 0 is given by x ( t) t 4t 6. (a) Find the velocity of the particle at any time t 0. (b) Find the acceleration of the particle at any time t 0. (c) When is the particle not moving? (d) For what values of t is the particle moving to the right? Moving to the left? (e) For what values of t is the particle slowing down? Speeding up?
Related Rates: The Process (1) Draw a diagram and LABEL the variables on the diagram () Write down the rates (derivatives) that you know and Write down the unknown rates (derivatives) that you want to find () Write an equation that that relates the variables (Use geometry, trigonometry and algebra to write the equation) Some problems require you to write TWO equations and then use substitution to combine these two equations into the ONE equation that relates the variables you need together. (4) Use the chain rule to take a derivative with respect to the control variable, time. (5) Substitute the given numbers into the resulting equation and solve for the unknown rate. Example 18: Water runs into a conical tank at a rate of 1.4 meters cubed per minute. The tank stands point down and has a height of 5 meters and a base diameter of 4 meters. How fast is the water level rising when the water is meters deep? Example 19: A woman 5.5 feet tall walks at a rate of feet per second away from a light that is 18 feet above the ground. (a) At what rate is the length of her shadow moving when she is 7 feet from the base of the light. (b) At what rate is the tip of her shadow changing when she is 7 feet from the base of the light.