Calculus AB Semester 1 Final Review

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1 Name Period Calculus AB Semester Final Review. Eponential functions: (A) kg. of a radioactive substance decay to kg. after years. Find how much remains after years. (B) Different isotopes of the same element can have very different half-lives. The decay of plutonium-.t is described by the formula Q= Qe, whereas the decay of plutonium- is described by.8t Q= Qe. Find the half-lives of plutonium- and plutonium-. (C) Suppose prices are increasing by.% per day. By what percent do prices increase in one year. Sketch the following graphs by hand (label two points on your graph): (A) y = (B) y = (C) y = ln (D) y = ln. Evaluate each limit. lim ln (B) lim( ln ) ( ln ) +. Evaluate each limit. (A) lim e ln (B) lim e. Evaluate each limit below, without using L hospital s Rule: (A) lim e (B) lim ln 6. Find the eact value of each epression: (A) log 8 (B) log6 (C) e ln 7 log+ log (D) (E) e ln

2 7. Determine whether f f = + (B) f = (C) f = cos( ) (D) f = + sin Calculus AB Semester Final Review Sheet is even, odd, or neither: 8. Sketch the following graphs by hand, and identify the amplitude and period of each: (A) y = cos (B) y = sin 9. The position of a car is given by the values in the table: t (seconds) s (feet) (A) Find the average velocity for the time period beginning when t = and lasting: sec., sec., sec. (B) Use the information from part (A) and other calculations to approimate the instantaneous velocity when t =.. Interpreting the derivative: (A) A yam has just been taken out of the oven and is cooling off before being eaten. The temperature, T, of the yam (measured in degrees Fahrenheit) is a function of how long it has been out of the oven, t (measured in minutes). Thus, we have T = f ( t). Is f ( t) positive or negative What are the units for f ( t) (B) An economist is interested in how the price of a certain commodity affects its sales. Suppose that at q = f p, eplain in economic a price of p dollars, a quantity q of the commodity is sold. If terms the meaning of the statements f =, and. Calculate derivative: y = (B) y = ( + ) (C) If f = ( ) 7. Calculate derivative: y = cos + (B) y = sin ( e ) (C) = ( + sin ), find f. y π. Find the equation of the line tangent to the given curve at the given point. π (A) y = tan, (, ) π (B) f = sin, =. Calculate derivative: y = csc sin (B) y = ln ( sec ). (C) If f ( ) =, find f f = 9,.

3 Calculus AB Semester Final Review Sheet. The table below gives values for the functions f and g, as well as their derivatives. - f g. f g. d (A) Find f ( ) g ( ) d d f d and d f d g d g f d (B) Find ( g ( )) and ( ) 6. Find dy d : (A) y + y = (B) sin cos at =. at =. a + by = y ( a and b are constants) (C) Find the slope of the curve + y = 7 at the point (-, ). 7. Derivatives and the graph of a function: Find all local ma./min. points and inflection points for y = ln, >. 8. Derivatives and the graph of a function: Find all local ma./min. points and inflection points for y= e. 9. Derivatives and the graph of a function: Find the absolute ma./min. values of y = on the interval [, ]. +. Linear approimation: (A) Find the linear approimation to the graph y = + near =. (B) Use linear approimation to estimate sin 9.. A continuous function defined for all has the following properties: f = f is increasing f is concave down f = (A) Sketch a possible graph of f. (B) How many zeros can f have (C) What can you say about the location of the zeros lim f (D) What is (E) Is it possible that f = (F) Is it possible that f =. Related Rates The volume of a cube is increasing at a rate of cm /min. How fast is the surface area increasing when the length of an edge is cm.. Related Rates A balloon is rising at a constant rate of ft./sec. A boy is cycling along a straight road at a speed of ft./sec. When he passes under the balloon it is ft above him. How fast is the distance between the boy and the balloon increasing sec. later

4 Calculus AB Semester Final Review Sheet. Related Rates The angle of elevation of the sun is decreasing at a rate of. rad./hr. How fast is the shadow cast by a -ft-tall building increasing when the angle of elevation is 6π. Related Rates A ft. ladder is leaning up against the side of a building, but slipping so that the top of the ladder is descending at a rate of ft./sec. What is the rate at which the base of the ladder moves away from the building when the top is 6 ft. from the ground 6. Optimization Find two positive integers such that the sum of the first number and four times the second number is and the product of the numbers is as large as possible. 7. Optimization The top and bottom margins of a poster are each 6 cm. and the side margins are each cm. If the area of printed material on the poster is 8 cm, find the dimensions of the entire poster with the smallest area. 8. L hospital s rule: sin ( a+ ) sin ( a+ ) + sin a (A) lim (B) lim ln e + π π tan 9. Riemann Sums (A) The value of ( ) d is estimated using subintervals. Find the estimates using left- and righthand sums. (B) Consider the function with values given in the table below. Approimate f right-hand Riemann sums with 6 subintervals.. Area under a curve: (A) Find the area under one arch of the curve y = sin. (B) Assume f ( ) is a positive function on the interval [,]. If ( ) under the curve f ( ) on the interval [,].. Area under a curve: (A) Find the area bounded by 6 f ( ) (B) Without a calculator, compute y = 9 and the -ais. d. 6 d using left- and f + d = 7, find the area

5 Calculus AB Semester Final Review Sheet. Average value of a function: (A) Assume the population, P, of Meico (in millions), is given by P = t, where t is the number of years since 98. What was the average population of Meico between 98 and 99 (B) A bar of metal is heated, and then allowed to cool. The temperature, H, of the bar t minutes after it.t starts cooling is given, in C, by H = + 98e. Find the average value of the temperature over the first hour.. Theorems about integrals: (A) Suppose f d = A and f d = B. What is f (B) Suppose g d =, find g 7 + d. 7 d (C) Suppose h( ) is an even function and h d = M, find h d.. A car is moving along a straight road from points A to B, starting from A at time t =. Below is the velocity (positive direction is from A to B ) plotted against time. (A) How many kilometers away from A is the car at time t =,,6,7,9 (B) What do you think is happening between t = 6 and t = 7 (C) Eplain what the car is doing at t = 7. Fundamental Theorem of Calculus (A) The rate at which the world s oil is being consumed is continuously increasing. Suppose the rate (in.t billions of barrels per year) is given by the function r = e, where t is measured in years since 99. Find the total quantity of oil used between 99 and 99. (B) Assume that r( t ) represents the rate at which a country s debt is growing, where t is years since 99. In terms of debt, eplain the meaning of r t dt.