Worksheets for MA 113

Transcription

1 Worksheet # : Review Worksheets for MA 3 Worksheet # 2: Functions, Logarithms, and Intro to Limits Worksheet # 3: Limits: A Numerical and Graphical Approach Worksheet # 4: Basic Limit Laws Worksheet # 5: Continuity Worksheet # 6: Algebraic Evaluation of Limits, Inverse Functions, and Trigonometric Functions Worksheet # 7: Trigonometric Functions and Limits Worksheet # 8: Review for Eam I Worksheet # 9: Limits at Infinity and Intermediate Value Theorem Worksheet # : Derivatives Worksheet # : Product and Quotient Rules Worksheet # 2: Higher Derivatives and Trigonometric Functions Worksheet # 3: Chain Rule Worksheet # 4: Implicit Differentiation and Inverse Functions Worksheet # 5: Related Rates of Change Worksheet # 6: Review for Eam II Worksheet # 7: Linear Approimation and Applications Worksheet # 8: Etreme Values and the Mean Value Theorem Worksheet # 9: The Shape of a Graph Worksheet # 2: L Hôpital s Rule & Optimization Worksheet # 2: Optimization Worksheet # 22: Newton s Method and Antiderivatives Worksheet # 23: Approimating Area Worksheet # 24: Review for Eam III Worksheet # 25: Definite Integrals and The Fundamental Theorem of Calculus Worksheet # 26: Net Change and The Substitution Method Worksheet # 27: Transcendental Functions and Other Integrals Worksheet # 28: Eponential Growth and Decay, and Area Between Curves Worksheet # 29: Review I for Final Worksheet # 3: Review II for Final 3 January 23

2 Worksheet # : Review. Find the equation of the line that passes through (, 2) and is parallel to the line 4 + 2y =. Put your answer in y = m + b form. 2. Find the slope, -intercept, and y-intercept of the line 3 2y = Write the equation of the line through (2, ) and (, 3) in point slope form. 4. Write the equation of the line containing (, ) and perpendicular to the line through (, ) and (2, 6). 5. The quadratic polynomial f() = 2 + b + c has roots at 3 and. What are the values of b and c? 6. Let f() = A 2 + B + C. If f() = 3, f( ) = 7, and f() = 4 what are the values of A, B and C? 7. Find the intersection of the lines y = 5 + and y = 8 3. Remember that an intersection is a point in the plane, hence an ordered pair. 8. Recall the definition of the absolute value function: { if = if <. Sketch the graph of this function. Also, sketch the graphs of the functions + 4 and A ball is thrown in the air from ground level. The height of the ball in meters at time t seconds is given by the function h(t) = 4.9t 2 + 3t. At what time does the ball hit the ground (be sure to use the proper units)?. We form a bo by removing squares of side length centimeters from the four corners of a rectangle of width cm and length 5 cm and then folding up the flaps between the squares that were removed. a) Write a function which gives the volume of the bo as a function of. b) Give the domain for this function.. True or False: (a) For any function f, f(s + t) = f(s) + f(t). (b) If f(s) = f(t), then s = t. (c) If s = t, then f(s) = f(t). (d) A circle can be the graph of a function. (e) A function is a rule which assigns eactly one output f() to every input. (f) If f() is increasing then f( 52.55) f(752.).

3 Worksheet # 2: Functions, Logarithms, and Intro to Limits. Let f() = 3 + and g() =. Find (f g)() and (g f)() and specify their domains Consider the function f() = 2. Find functions g() and h() so that f() can be written as + 3 f() = (g h)(). 3. Suppose the graph of g() is given by the equation g() = f(2 5) + 7. In terms of standard transformations describe how to obtain the graph of g() from the graph of f(). 4. Find the domain and range of the following functions. (a) f() = 5 (b) f() = Compute each of the following logarithms eactly. Do not use a calculator. (a) log 3 (/27) (b) log 2 (6) log 2 (5) + log 2 (2) (c) log (log (log ( ))) 6. Solve the following equations for : (a) 2+ 7 = (b) log 2 () + log 2 ( ) = 7. Sketch the graphs of the following functions using your knowledge of basic functions and transformations. Then sketch the tangent line to the curve at the specified point. (a) f() = + 3, = (b) f() = ( 2) 3, = 2 8. A particle is moving along a straight line so that its position at time t seconds is given by s(t) = 4t 2 t. (a) Find the average velocity of the particle over the time interval [, 2]. (b) Determine the average velocity of the particle over the time interval [2, t] where t > 2. Simplify your answer. [Hint: Factor the numerator.] (c) Based on your answer in (b) can you guess a value for the instantaneous velocity of the particle at t = 2? 9. Let s(t) be the function which describes the position of a particle traveling along the y-ais. Suppose the point (5, 6) is on the graph y = s(t) (in the t-y plane) and the tangent line at this point is given by y = 6. At time t = 5, determine the particle s position and instantaneous velocity.. The point P (3, ) lies on the curve y = 2. (a) If Q is the point (, 2), find a formula for the slope of the secant line P Q. (b) Using your formula from part (a) and a calculator, find the slope of the secant line P Q for the following values of (do not round until you get to the final answer): 2.9, 2.99, 2.999, 3., 3., and 3. TI-8 Calculator Tip: Enter the formula under y= and then use Table. (c) Using the results of part (b), guess the value of the slope of the tangent line to the curve at P (3, ).

4 (d) Using the slope from part (c), find the equation of the tangent line to the curve at P (3, ).. True or False: (a) The graph of every function will pass the vertical line test. (b) f g() = g f(). (c) There is a function whose graph is an oval. (d) log 3 (3 ) = for all.

5 Worksheet # 3: Limits: A Numerical and Graphical Approach. Comprehension check: (a) In words, describe what a f() = L means. (b) In words, what does a f() = mean? (c) Suppose f() = 2. Does f() = 2? (d) Suppose f() = 2. Does f() = 2? 2. Compute the value of the following functions near the given value. Use this information to guess the value of the it of the function (if it eist) as approaches the given value. (a) f() = ( 2) 3, = (b) f() = , = 3 2 (c) f() =, = (d) f() = 2 +, = (e) f() = , = 2 2 if 3. Let f() = if < and 2. 3 if = 2 (a) Sketch the graph of f. (b) Compute the following: i. f() ii. f() + iii. f() iv. f() v. f() 2 vi. f() 2 + vii. f() 2 viii. f(2) 4. In the following, sketch the functions and use the sketch to compute the it. (a) π 3 (b) π (c) a (d) Compute the following its or eplain why they fail to eist: (a) (b) (c)

6 (d) 3 6. In the theory of relativity, the mass of a particle with velocity v is: m = m v2 c 2 where m is the mass of the particle at rest and c is the speed of light. What happens as v c? if > 2 7. Let f() = a if = 2. Find k and a so that f() = f( 2). 2 k if < 2

7 Worksheet # 4: Basic Limit Laws. Given f() = 5 and g() = 2, use it laws to compute the following its or eplain why 2 2 we cannot find the it. Note when working through a it problem that your answers should be a chain of true equalities. Make sure to keep the operator until the very last step. a (a) 2 (2f() g()) (b) 2 (f()g(2)) (c) 2 f()g() (d) 2 f() 2 + g() 2 (e) 2 [f()] 3 2 (f) 2 f() 5 g() 2 2. Calculate the following its if they eist or eplain why the it does not eist. (a) 2 (b) 2 2 (c) (d) a (a a 2 ) (e) a (a a 2 ) 3. Can the quotient law of its be applied to evaluate sin? 4. If h 3 (hf(h)) = 5, can you find h 3 f(h)? Use the it laws to justify your answer c 5. Find the value of c such that eists. What is the it? 2 2 { 2 + c if 5 6. Let g() = 3 if > 5 Find the value of c such that 5 g() eists. h h 7. Show that does not eist by eamining one-sided its. Then sketch the graph of h h h and check your reasoning. 8. True or False: ( + 2)( ) (a) Let f() = and g() = + 2. Then f() = g(). ( + 2)( ) (b) Let f() = and g() = + 2. Then f() = g(). (c) If both the one-sided its of f() eist as approaches a, then f() eists. a (d) If f() eists then f() = f(a). a a 9. Draw a graph of two functions f() and g() such that (f() + g()) eist but neither f() nor g() eist.

8 Worksheet # 5: Continuity. Comprehension check: (a) Define what it means for f() to be continuous at the point = a. What does it mean if f() is continuous on the interval [a, b]? What does it mean to say f() is continuous? (b) There are three distinct ways in which a function will fail to be continuous at a point = a. Describe the three types of discontinuity. Provide a sketch and an eample of each type. (c) True or false? Every function is continuous on its domain. (d) True or false? The sum, difference, and product of continuous functions are all continuous. (e) If f() is continuous at = a, what can you say about f()? a + f()g() f() 3 (f) Suppose f(), g() are continuous everywhere. What is a g() 2? + 2. Using the definition of continuity and properties of its, show that the following functions are continuous at the given point a. (a) f() = π, a = (b) f() = , a = + 3 (c) f() = 2 9, a = 4 3. Give the largest domain on which the following functions are continuous. Use interval notation. + (a) f() = (b) f() = 2 + (c) f() = if (d) f() = + if < < 2 ( 2) 2 if 2 c 2 5 if < 4. Let c be a number and consider the function f() = if =. 2c if > (a) Find all numbers c such that f() eists. (b) Is there a number c such that f() is continuous at =? Justify your answer if 4 5. Find parameters a and b so that f() = a + b if 4 < < 3 is continuous if 3 6. Suppose that f() and g() are continuous functions where f(2) = 5 and g(6) =. Compute the following: [f()] 2 + (a)

9 g() + 4 (b) 6 f ( ) 3 g() { 6 6 for 6, 7. Suppose that: f() = for = 6 Determine the points at which the function f() is discontinuous and state the type of discontinuity.

10 Worksheet # 6: Algebraic Evaluation of Limits, Inverse Functions, and Trigonometric Functions. Let f() = f( + h) f() (a) Let g() = and find g(). h h (b) What is the geometric meaning of g(4)? (c) What is the domain of g()? 2. For each it, evaluate the it or or eplain why it does not eist. Use the it rules to justify each step. It is good practice to sketch a graph to check your answers. + 2 (a) (b) (c) (d) 2 ( (e) ) (f) (2 + ) Consider the function f() = + ln(). Determine the inverse function of f. 4. Consider the function whose graph appears below. y=f() (a) Find f(3), f (2) and (f(2)). (b) Give the domain and range of f and of f. (c) Sketch the graph of f. 5. Convert the angle π/2 to degrees and the angle 9 to radians. Give eact answers. 6. Find the eact values of the following epressions. Do not use a calculator. (a) tan () (b) tan(tan ()) (c) sin (sin(7π/3)) (d) tan(sin (.8)) 7. Let O be the center of a circle whose circumference is 48 centimeters. Let P and Q be two points on the circle that are endpoints of an arc that is 6 centimeters long. Find the angle between the segments OQ and OP. Epress your answer in radians. Find the distance between P and Q. 8. If π/2 θ 3π/2 and tan θ = 4/3, find sin θ, cos θ, cot θ, sec θ, and csc θ. 9. Find all solutions to the following equations in the interval [, 2π]. You will need to use some trigonometric identities.

11 (a) 3 cos() + 2 tan() cos 2 () = (b) 3 cot 2 () = (c) 2 cos() + sin(2) =. True or False: (a) Let p() = c n n + c n n c + c be a polynomial with coefficients c n, c n,..., c. Then a p() = c n a n + c n a n c a + c. (b) If (f()) 2 eists, then f() eists. (c) Every function has an inverse. (d) The graph of every function will pass the horizontal line test.

12 Worksheet # 7: Trigonometric Functions and Limits The tetbook uses both sin and arcsin to denote the inverse of the sin function restricted to the interval [ π/2, π/2]. This worksheet uses the arc prefi for the inverse trigonometric functions.. Let α π 2 and cos (α) = 5 3. Find sin(α), tan(α), cot(α), sec(α), and csc(α). 2. Evaluate (without using your calculator): sin(arcsin( 3/5)), cos(arcsin( 3/5)), and tan(arcsin( 3/5)). 3. Suppose that π β 3π/2 and tan(β) = 3. Find cos(β), sin(β), and cos(β + π). What is β? 4. Let f() = sin() for satisfying π/2 3π/2. Sketch the graphs of f and f. 5. Evaluate arccos(cos(5π/2)). 6. Find the values of for which each of the following equations are true. Can you find values of for which one of the equations fails to hold true? (a) cos(arccos()) = (b) arcsin(sin()) =. 7. Let f() = + 2 sin(/) for. Find two simpler functions g and h so that we can use the squeeze theorem to show f() = g() = h(). Give the common value of the its. Use your calculator to produce a graph that illustrates that the squeeze theorem applies. 8. Evaluate the its sin(2t) (a) t t tan(2t) (b) t t cos(t) tan(2t) (c) t t (d) csc(3t) tan(2t) t (e) t cos(2t) t (f) t cos(2t) sin 2 (3t) (g) cos(t) t π/2 t Hint: Multiply by cos(2t). The following identity may be useful for the net problems. 9. Use equation (), to simplify the it t 2. (a) Evaluate the it t sin t. cos( + y) = cos() cos(y) sin() sin(y) () cos( + h) cos() h h cos(5t) cos 2 (5t) (b) Find the it. t t tan() (c) Evaluate the it. 5 cos(2) (d) Evaluate the it Hint: Use equation () cos() cos(3) (e) Evaluate the it. Hint: Multiply and divide by + cos(3) 2 (f) Evaluate the it cos cos 3 2. Hint: Use equation () to rewrite cos(3) as cos( + 2)

13 Worksheet # 8: Review for Eam I. Find all real numbers of the constant a and b for which the function f() = a + b satisfies: (a) f f() = f() for all. (b) f f() = for all. 2. Simplify the following epressions. (a) log 5 25 (b) (log 4 6)(log 4 2) (c) log log 5 3 (d) log ((log y y )) (e) log π ( cos ) + log π ( + cos ) 2 log π sin 3. (a) Solve the equation = 4 for. Show each step in the computation. (b) Epress the quantitiy log 2 ( 3 2) + 3 log 2() log 2 (5) as a single logarithm. 4. Calculate the following its using the it laws. Carefully show your work and use only one it law per step. (a) (2 ) (b) 5. Calculate the following its if they eist or eplain why the it does not eist. 2 (a) 2 2 (b) (c) 2 (d) (e) sin() 6. Use the fact that = to find sin(3). 7. (a) State the Squeeze Theorem. (b) Use the Squeeze Theorem to find the it sin 2 8. Use the Squeeze Theorem to find cos cos(tan ) π 2 9. If f() = 3 2, find f(), f() and f() (a) State the definition of the continuity of a function f() at = a.

14 (b) Find the constant a so that the function is continuous on the entire real line. 2 a 2 if a f() = a 8 if = a. Complete the following statements: (a) A function f() passes the horizontal line test, if the function f is... (b) If a f() and a g() eist, then (c) a f() f() a g() = a g() provided... a f() = f() = L if and only if... + a { if 2 (d) Let g() = be a piecewise function. if = 2 The function g() is NOT continuous at = 2 since... 2 if < (e) Let f() = if = be a piecewise function. if > The function f() is NOT continuous at = since...

15 Worksheet # 9: Limits at Infinity and Intermediate Value Theorem. (a) Describe the behavior of the function f() if f() = L and f() = M. (b) Eplain the difference between f() = and f() = Evaluate the following its, or eplain why the it does not eist: (a) (b) (c) (d) (e) (f) ± Find the its f() and ( 2 f() if f() = + 4. Sketch a graph with all of the following properties: ) 2. f(t) = 2 t f(t) = t f(t) = t + t f(t) = t 4 f(t) = 3 f(4) = 6 5. Find the following its; (a) (b) (c) sin cos 6. (a) State the Intermediate Value Theorem. (b) Show that f() = 3 + has a zero in the interval [, ]. 7. Use the Intermediate Value Theorem to find an interval of length in which a solution to the equation = 5 must eist. 8. Show that there is some a with < a < 2 such that a 2 + cos(πa) = Show that the equation ln() = e has a solution between and 2. { if. Let f() = be a piecewise function. if > Although f( ) = and f() =, f() /2 for all in its domain. Why doesn t this contradict to the Intermediate Value Theorem?. Prove that 4 = has no solution.

16 Worksheet # : Derivatives. Comprehension check: (a) Are differentiable functions also continuous? Are continuous functions also differentiable? (b) What does the derivative of f() at = a describe graphically? (c) True or false: If f () = g () then f() = g()? (d) True or false: (f() + g()) = f () + g () (e) How is the number e defined? 2. Consider the graph below of the function f() on the interval [, 5]. (a) For which values would the derivative f () not be defined? (b) Sketch the graph of the derivative function f. 3. Find f (a) using either form of the definition for the derivative: 4. Let (a) f() = , a = 2. (b) f() = + 3, a =. (c) f() =, a = 9. h(t) = Find a and b so that h is differentiable at t =. 5. Compute the derivative of the following functions: { at + b if t t 3 + if t >. (a) f() = (b) h() = 3e + 2 +

17 (c) k() = A 4 + B2 + C + D ( (d) l() = + ) 2 6. Find an equation for the tangent line to the curve y = 3/2 + 2 at = Find the equation of each tangent line to the parabola y = 2 which pass through the point (, ). First sketch the graph of the parabola and the desired tangent line(s). 8. Consider the function f() = Use the Intermediate Value Theorem to show that the graph of f has a horizontal tangent line between = 3 and = Find a function f and a number a so that the following it represents a derivative f (a). h (4 + h) 3 64 h

18 Worksheet # : Product and Quotient Rules. Show by way of eample that, in general, and d df (f g) d d dg d d d ( ) f g df d. dg d 2. Calculate the derivatives of the following functions in the two ways that are described. (a) f(r) = r 3 /3 i. using the constant multiple rule and the power rule ii. using the quotient rule and the power rule Which method should we prefer? (b) f() = 5 i. using the power rule ii. using the product rule by considering the function as f() = 2 3 (c) g() = ( 2 + )( 4 ) i. first multiply out the factors and then use the power rule ii. by using the product rule 3. State the quotient and product rule and be sure to include all necessary hypotheses. 4. Compute the first derivative of each of the following: (a) f() = (e) f() = (b) f() = (3 2 + )e (c) f() = e 2 3 (d) f() = ( e ) ( ) (f) f() = a + b c + d (g) f() = (2 + )( 3 + 2) 5 (h) f() = ( 3)(2 + )( + 5) 5. Let f() = (3 )e. For which is the slope of the tangent line to f positive? Negative? Zero? 6. Find an equation of the tangent line to the given curve at the specified point. Sketch the curve and the tangent line to check your answer. e (a) y = at the point = 3. + (b) y = 2e at the point =. 7. Suppose that f(2) = 3, g(2) = 2, f (2) = 2, and g (2) = 4. For the following functions, find h (2). (a) h() = 5f() + 2g() (b) h() = f()g() (c) h() = f() g() (d) h() = g() + f()

19 Worksheet # 2: Higher Derivatives and Trigonometric Functions. Given that G(t) = 4t 2 3t + 42 find the instantaneous rate of change when t = An object is thrown upward with an initial velocity of 5 m/s from an initial height of 4 m. Find the velocity of the object when it hits the ground. Assume that the acceleration of gravity is 9.8 m/s Suppose that height of a triangle is equal to its base b. Find the instantaneous rate of change in the area respect to the base b when the base is Calculate the indicated derivative: (a) f (4) (), f() = 4 (b) g (3) (5), g() = (c) h (3) (t), h(t) = 4e t t 3 (d) s (2) (w), s(w) = we w 5. Calculate the first three derivatives of f() = e and use these to guess a general formula for f (n) (), the n-th derivative of f. 6. Differentiate each of the following functions: (a) f(t) = cos(t) (b) g(u) = cos(u) (c) r(θ) = θ 3 sin(θ) (d) s(t) = tan(t) + csc(t) (e) h() = sin() csc() (f) f() = 2 sin 2 () (g) g() = sec() + cot() 7. Calculate the first five derivatives of f() = sin(). Then determine f (8) and f (37) 8. Calculate the first 5 derivatives of f() = /. Can you guess a formula for the nth derivative, f (n)? 9. A particle s distance from the origin (in meters) along the -ais is modeled by p(t) = 2 sin(t) cos(t), where t is measured in seconds. (a) Determine the particle s speed (speed is defined as the absolute value of velocity) at π seconds. (b) Is the particle moving towards or away from the origin at π seconds? Eplain. (c) Now, find the velocity of the particle at time t = 3π 2 away from the origin? (d) Is the particle speeding up at π 2 seconds?. Is the particle moving toward the origin or. Find an equation of the tangent line at the point specified: (a) y = 3 + cos(), = (b) y = csc() cot(), = π 4 (c) y = e θ sec(θ), θ = π 4

20 Worksheet # 3: Chain Rule. (a) Carefully state the chain rule using complete sentences. (b) Suppose f and g are differentiable functions so that f(2) = 3, f (2) =, g(2) = 4, and g (2) = 2. Find each of the following: i. h (2) where h() = [f()] ii. l (2) where l() = f( 3 g()). 2. Given the following functions: f() = sec(), and g() = Find: (a) f(g()) = (b) f () = (c) g () = (d) f (g()) = (e) (f g) () = 3. Differentiate each of the following and simplify your answer. (a) f() = (b) g(t) = tan(sin(t)) (c) h(u) = sec 2 (u) + tan 2 (u) (d) f() = e (32 +) (e) g() = sin(sin(sin())) 4. Find an equation of the tangent line to the curve at the given point. (a) f() = 2 e 3, = 2 (b) f() = sin() + sin 2 (), = 5. Compute the derivative of 2 + (a) Using the quotient rule. (b) Rewrite the function in two ways: Check that both answers give the same result. 2 + = (2 + ) and use the product and chain rule. 6. If h() = 4 + 3f() where f() = 7 and f () = 4, find h (). 7. Let h() = f g() and k() = g f() where some values of f and g are given by the table Find: h ( ), h (3) and k (2). f() g() f () g () Find all values so that f() = 2 sin() + sin 2 () has a horizontal tangent at. 9. Comprehension check for derivatives of trigonometric functions:

21 (a) True or False: If f (θ) = sin(θ), then f(θ) = cos(θ). (b) True or False: If θ is one of the non-right angles in a right triangle and sin(θ) = 2, then the 3 hypotenuse of the triangle must have length 3. (c) Differentiate both sides of the double-angle formula for the cosine function, cos(2) = cos 2 () sin 2 (). Do you obtain a familiar identity?

22 Worksheet # 4: Implicit Differentiation and Inverse Functions. Find the derivative of y with respect to : (a) ln (y) = cos(y 4 ). (b) y 2 3 = π 2 3. ( (c) sin(y) = ln y ). 2. Consider the ellipse given by the equation ( 2) (y 3)2 8 =. (a) Find the equation of the tangent line to the ellipse at the point (u, v) where u = 4 and v >. (b) Sketch the ellipse and the line to check your answer. 3. Find the derivative of f() = π tan (ω), where ω is a constant. 4. Let (a, b) be a point in the circle 2 + y 2 = 44. Use implicit differentiation to find the slope of the tangent line to the circle at (a, b). 5. Let f() be an invertible function such that g() = f (), f(3) = 5 and f (3) = 2. Using only this information find the equation of the tangent line to g() at = Let y = f() be the unique function satisfying 2 + = 4. Find the slope of the tangent line to f() 3y at the point ( 2, 9 ). 7. The equation of the tangent line to f() at the point (2, f(2)) is given by the equation y = If G() = 4f(), find G (2). 8. Differentiate both sides of the equation, V = 4 3 πr3, with respect to V and find dr dv when r = 8 π. 9. Use implicit differentiation to find the derivative of arctan(). Thus if = tan(y), use implicit differentiation to compute dy/d. Can you simplify to epress dy/d in terms of?. (a) Compute d d arcsin(cos()). (b) Compute d d (arcsin() + arccos()). Give a geometric eplanation as to why the answer is. (c) Compute d ( ( ) ) tan + tan () and simplify to show that the derivative is. Give a d geometric eplanation of your result.. Consider the line through (, b) and (2, ). Let θ be the directed angle from the -ais to this line so that θ > when b <. Find the derivative of θ with respect to b. 2. Let f be defined by f() = e 2. (a) For which values of is f () = (b) For which values of is f () = 3. The notation tan () is ambiguous. It is not clear if the eponent indicates the reciprocal or the inverse function. If we allow both interpretations, how many different ways can you (correctly) compute the derivative f () for f() = (tan ) ()? In order to avoid this ambiguity, we will generally use cot() for the reciprocal of tan() and arctan() for the inverse of the tangent function restricted to the domain ( π/2, π/2).

23 Worksheet # 5: Related Rates of Change. Let a and b denote the length in meters of the two legs of a right triangle. At time t =, a = 2 and b = 2. If a is decreasing at a constant rate of 2 meters per second and b is increasing at a constant rate of 3 meters per second. Find the rate of change of the area of the triangle at time t = 5 seconds. 2. A person 6 feet tall walks along a straight path at a rate of 4 feet per second away from a streetlight that is 5 feet above the ground. Assume that at time t = the person is touching the streetlight. (a) Draw a picture to represent the situation. (b) Find an equation that relates the length of the person s shadow to the person s position (relative to the streetlight). (c) Find the rate of change in the length of the shadow when t = 3. (d) Find how fast is the tip of the person s shadow is moving when t = 4. (e) Does the precise time make a difference in these calculations? 3. An spherical snow ball is melting. The rate of change of the surface area of the snow ball is constant and equal to 7 square centimeters per hour. Find the rate of change of the radius of the snow ball when r = 5 centimeters. 4. The height of a cylinder is a linear function of its radius (i.e. h = ar + b for some a, b constants). The height increases twice as fast as the radius r and dr dt is constant. At time t = seconds the radius is r = feet, the height is h = 3 feet and the rate of change of the volume is 6π cubic feet/second. (a) Find an equation to relate the height and radius of the cylinder. (b) Epress the volume as a function of the radius. (c) Find the rate of change of the volume when the radius is 4 feet. 5. A water tank is shaped like a cone with the verte pointing down. The height of the tank is feet and diameter of the base 5 feet. At time t = the tank is full and starts to be emptied. After 3 minutes the height of the water is 4 feet and it is decreasing at a rate of feet per minute. At this instant, find the rate of change of the volume of the water in the tank (recall that the volume of a cone is given by V = 3 πr2 h). (Give the units for your answer) 6. A car moves at 5 miles per hour on a straight road. A house is 2 miles away from the road. What is the rate of change in the angle between the house and the car and the house and the road when the car passes the house. 7. A car moves along a road that is shaped like the parabola y = 2. At what point on the parabola are the rates of change for the and y coordinates equal? 8. Let f() = + 3 and h() = + f() (a) Find f (). (b) Use the previous result to find h (). (c) Let = (t) be a function of time t with () = and set F (t) = h((t)). If F () = 8, find () given that F () = 8.

24 . Evaluate the following its; Worksheet # 6: Review for Eam II (a) (b) (c) (d) If g() = , use the Intermediate Value Theorem to show that there is a number a such that g(a) =. 3. (a) State the definition of the derivative of a function f() at a point a. (b) Find a function f and a number a such that f() f(a) a (c) Evaluate the following it by using (a) and (b), = ln(2 ) ln(2 ) 4. State the following rules with the hypotheses and conclusion. (a) The product rule and quotient rule. (b) The chain rule. 5. A particle is moving along a line so that at time t seconds, the particle is s(t) = 3 t3 t 2 8t meters to the right of the origin. (a) Find the time interval(s) when the particle is moving to the left. (b) Find the time(s) when the velocity is zero. (c) Find the time interval(s) when the particle s velocity is increasing. (d) Find the time interval(s) when the particle is speeding up. 6. Compute the first derivative of each of the following functions: (a) f() = cos(4π 3 ) + sin(3 + 2) (b) b() = 4 cos(3 2 ) (c) y(θ) = e sec(2θ) (d) k() = ln(7 2 + sin() + ) (e) u() = (sin (2)) 2 (f) h() = cos(2) (g) m() = (h) q() = e + 2 (i) n() = cos (tan()) (j) w() = arcsin() arccos() 7. Let f() = cos(2). Find the fourth derivative at =, f (4) (). 8. Let f be a one to one, differentiable function such that f() = 2, f(2) = 3, f () = 4 and f (2) = 5. Find the derivative of the inverse function, (f ) (2).

25 9. The tangent line to f() at = 3 is given by y = 2 4. Find the tangent line to g() = f() at = 3. Put your answer in slope-intercept form.. Consider the curve y y 2 = 24. Assume this equation can be used to define y as a function of (i.e. y = y()) near (, 2) with y() = 2. Find the equation of the tangent line to this curve at (, 2). ( π. Let be the angle in the interval 2, 3π ) so that sin() = 3. Find: sin( ), cos(), and cot() Each side of a square is increasing at a rate of 5 cm/s. At what rate is the area of the square increasing when the area is 4 cm 2? 3. The sides of a rectangle are varying in such a way that the area is constant. At a certain instant the length of a rectangle is 6 m, the width is 2 m and the width is increasing at 3 m/s. What is the rate of change of the length at this instant? 4. Suppose f and g are differentiable functions such that f(2) = 3, f (2) =, g(2) = 4, and g (2) = 2. Find: (a) h (2) where h() = ln([f()] 2 ); (b) l (2) where l() = f( 3 g()). 5. Abby is driving north along Ash Road. Boris driving west on Birch Road. At :57 am, Boris is 5 km east of Oakville and traveling west at a speed of 6 km/h and Abby is km north of Oakville and traveling north at a speed of 5 km/h. (a) Make a sketch showing the location and direction of travel for Abby and Boris. (b) Find the rate of change of the distance between Abby and Boris at :57 AM. (c) At :57 AM, is the distance between Abby and Boris increasing, decreasing, or not changing?

26 Worksheet # 7: Linear Approimation and Applications. For each of the following, use a linear approimation to the change in the function and a convenient nearby point to estimate the value: (a) (3.) 3 (b) 7 (c) 8.6 2/3 (d) tan(44 ) 2. What is the relation between the linearization of a function f() at = a and the tangent line to the graph of the function f() at = a on the graph? 3. Use the linearization of at = 6 to estimate 8; (a) Find a decimal approimation to 8 using a calculator. (b) Compute both the error and the percentage error. 4. Suppose we want to paint a sphere of radius 2 cm with a coat of paint.2 cm thick. Use a linear approimation to approimate the amount of paint we need to do the job. 5. Let f() = 6 +. First, find the linearization to f() at =, then use the linearization to estimate Present your solution as a rational number. 6. Find the linearization L() to the function f() = 2 at = Find the linearization L() to the function f() = at = 4, then use the linearization to estimate Your physics professor tells you that you can replace sin(θ) with θ when θ is close to zero. Eplain why this is reasonable. 9. Suppose we measure the radius of a sphere as cm with an accuracy of ±.5 cm. Use linear approimations to estimate the maimum error in: (a) the computed surface area. (b) the computed volume.. Suppose that y = y() is a differentiable function which is defined near = 2, satisfies y(2) = and 2 + 3y 2 + y 3 = 9. Use the linear approimation to the change in y to approimate the value of y(.9).

27 Worksheet # 8: Etreme Values and the Mean Value Theorem. (a) Define the following terms or concepts: Critical point f has a local maimum at = a Absolute maimum (b) State the following: The First Derivative Test for Critical Points The Mean Value Theorem 2. Sketch the following: (a) The graph of a function defined on (, ) with three local maima, two local minima, and no absolute minima. (b) The graph of a continuous function with a local maimum at = but which is not differentiable at =. (c) The graph of a function on [, ) which has a local maimum but not an absolute maimum. (d) The graph of a function on [, ] which has a local maimum but not an absolute maimum. (e) The graph of a discontinuous function defined on [, ] which has both an absolute minimum and absolute maimum. 3. Find the critical points for the following functions: (a) f() = (b) g() = e 3 ( 2 7) (c) h() = 5 4. Find the absolute maimum and absolute minimum values of the following functions on the given intervals. Specify the -values where these etrema occur. (a) f() = , [ 2, 3] (b) h() = + 2, [, ] 5. (a) Consider the function f() = on (, ). i. Find the critical point(s) of f(). ii. Find the intervals on which f() is increasing or decreasing. iii. Find the local etrema of f(). (b) Repeat with the function f() = on (, ). [ π 2, π ]. 2 (c) Repeat with the function f() = sin 2 () cos() on 6. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. (a) f() = on the interval [, 4] + 2 (b) f() = sin() cos() on the interval [, 2π] 7. Comprehension check: (a) True or False: If f (c) = then f has a local maimum or local minimum at c. (b) True or False: If f is differentiable and has a local maimum or minimum at = c then f (c) =. (c) A function continuous on an open interval may not have an absolute minimum or absolute maimum on that interval. Give an eample of continuous function on (, ) which has no absolute maimum. (d) True or False: If f is differentiable on the open interval (a, b), continuous on the closed interval [a, b], and f () for all in (a, b), then we have f(a) f(b).

28 Worksheet # 9: The Shape of a Graph. Eplain how to use the second derivative test to identify and classify local etrema of a twice differentiable function f(). Does the test always work? What should you do if it fails? 2. Suppose that g() is differentiable for all and that 5 g () 2 for all. Assume also that g() = 2. Based on this information, use the Mean Value Theorem to determine the largest and smallest possible values for g(2). 3. A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 59 miles on a toll road with speed it 65 mph. The trucker was cited for speeding. Why did she deserve the ticket? 4. (a) Consider the function f() = i. Find the intervals on which the graph of f() is increasing or decreasing. ii. Find the intervals of concavity of f(). iii. Find the points of inflection of f(). (b) Repeat with the function f() = 2 + sin() on ( π 2, 3π ). 2 (c) Repeat with the function f() = For each of the following graphs: (a) Find the intervals where the function is increasing and decreasing respectively. (b) Find the intervals of concavity for each function. (c) Identify all local etrema and inflection points on the interval (,6). 6. Find the local etrema of the following functions using the second derivative test: (a) f() = (b) g() = 5 ln(2) 7. Find the local etrema of f() = using the second derivative where possible. 8. Sketch a graph of a continuous function f() with the following properties: f is increasing on (, 3) (, 7) (7, ) f is decreasing on ( 3, ) f is concave up on (, 3) (7, ) f is concave down on (, ) (3, 7)

29 Worksheet # 2: L Hôpital s Rule & Optimization. Carefully state l Hôpital s Rule. 2. Compute the following its. Use l Hôpital s Rule where appropriate but first check that no easier method will solve the problem. 9 (a) 5 sin(4) (b) tan(5) (c) (d) Find the dimensions of and y of the rectangle of maimum area that can be formed using 3 meters of wire. (a) What is the constraint equation relating and y? (b) Find a formula for the area in terms of alone. (c) Solve the optimization problem. 4. A fleible tube of length 4 m is bent into an L-shape. Where should the bend be made to minimize the distance between the two ends? 5. A rancher will use 6 m of fencing to build a corral in the shape of a semicircle on top of a rectangle. Find the dimensions that maimize the area of the corral. (Hint: draw a picture) 6. Find the value A for which we can use l Hôpital s rule to evaluate the it 2 For this value of A, give the value of the it. 2 + A Compute the following its. Use l Hôpital s Rule where appropriate but first check that no easier method will solve the problem. (a) (b) 2 e 3 e 2 (c) π cos() + 2 π 2 (d) (arctan() π 2 ) 8. Find the dimensions and y of the rectangle inscribed in a circle of radius r that maimizes the quantity y Find the point on the line y = closest to the point (, ). Find the point on the line y = closest to the point (r, r). What do these points look like graphically?

30 Worksheet # 2: Optimization. Suppose that f is a function on an open interval I = (a, b) and c is in I. Suppose that f is continuous at c, f () > for > c and f () < c for < c. Is f(c) an absolute minimum value for f on I? Justify your answer. 2. Find the point(s) on the hyperbola y = 6 that is (are) closest to (, ). Be sure to clearly state what function you choose to minimize or maimize and why. 3. A farmer has 24 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? 4. A hockey team plays in an arena with a seating capacity of 5 spectators. With the ticket price set at $2, average attendance at a game has been. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by. How should the owners of the team set the ticket price to maimize their revenue from ticket sales? 5. An oil company needs to run a pipeline to a nearby station. The station and oil company are on opposite sides of a river that is km wide, and that runs eactly west-east. Also, the station is km east of where the oil company would be if it was on the same side of the river. The cost of land pipe is $2 per meter and the cost of water pipe is $3 per meter. Set up an equation whose solution(s) are the critical points of the cost function for this problem. 6. A ft length of rope is to be cut into two pieces to form a square and a circle. How should the rope be cut to maimize the enclosed area? 7. Consider a can in the shape of a right circular cylinder. The top and bottom of the can is made of a material that costs 4 cents per square centimeter, and the side is made of a material that costs 3 cents per square centimeter. We want to find the dimensions of the can which has volume 72 π cubic centimeters, and whose cost is as small as possible. (a) Find a function f(r) which gives the cost of the can in terms of radius r. Be sure to specify the domain. (b) Give the radius and height of the can with least cost. (c) Eplain how you known you have found the can of least cost. 8. A bo is to have a square base, no top, and a volume of cubic centimeters. What are the dimensions of the bo with the smallest possible total surface area? Provide an eact answer; do not convert your answer to decimal form. Make a sketch and introduce all the notation you are using.

31 Worksheet # 22: Newton s Method and Antiderivatives. Use Newton s method to find an approimation to 3 2. You may do this by finding a solution of 3 2 =. 2. Use Newton s method to approimate the critical points of the function f() = Let f() = + 2. (a) Solve f() = without using Newton s method. (b) Use Newton s method to solve f() = beginning with the starting point = 2. Does something interesting happen? (c) Make a sketch of the graph of f and eplain what you observed in part b). 4. (a) Let f() = Calculate the derivative f (). What is an anti-derivative of f ()? (b) Let g() = 2 +. Let G() be any anti-derivative of g. What is G ()? 5. Find f given that 6. Find g given that f () = sin() sec() tan(), f(π) =. g (t) = 9.8, g () =, g() = 2. On the surface of the earth, the acceleration of an object due to gravity is approimately 9.8 m/s 2. What situation could we describe using the functino g? Be sure to specify what g and its first two derivatives represent. 7. A small rock is dropped from a bridge and the splash is heard 3 seconds later. How high is the bridge? 8. Let f be a function on the domain (, ) that satisfies (f ) 2 =. This is an eample of a differential equation. Suppose also that we are given an initial value condition f() =. (a) Show that this does not have a unique solution by finding two different functions that satisfy both conditions. (b) What does the fact that there are multiple solutions say about this as a model for physical phenomena? 9. Find a function f() such that f () = f(). Find the solution, given initial condition f() = π.. Let f() = /, F () = ln( ), and ln(), > G() = ln( ) + 8, <. (a) Is F an anti-derivative of f? Is G an anti-derivative of f? Is F G equal to a constant? (b) Does Theorem on page 275 imply that F G is constant? Is the theorem wrong?

32 Worksheet # 23: Approimating Area. Write each of following in summation notation: (a) (b) (c) Compute (i + j). i= j= The following summation formulas will be useful below. n n(n + ) n j =, j 2 = 2 j= 3. Find the number n such that n i = 78. i= 4. Give the value of the following sums. (a) (b) 2 j= 2 j= (2k 2 + 3) (3k + 2) j= n(n + )(2n + ) 6 5. The velocity of a train at several times is shown in the table below. Assume that the velocity changes linearly between each time given. (a) Plot the velocity of the train versus time. t=time in minutes v(t)=velocity in Km/h (b) Compute the left and right-endpoint approimations to the area under the graph of v. (c) Eplain why these approimate areas are also an approimation to the distance that the train travels. 6. Let f() = /. Divide the interval [, 3] into five subintervals of equal length and compute R 5 and L 5, the left and right endpoint approimations to the area under the graph of f in the interval [, 3]. Is R 5 larger or smaller than the true area? Is L 5 larger or smaller than the true area? 7. Let f() = 2. Divide the interval [, ] into four equal subintervals and compute L 4 and R 4, the left and right-endpoint approimations to the area under the graph of f. Is R 4 larger or smaller than the true area? Is L 4 larger or smaller than the true area? What can you conclude about the value π? 8. Let f() = 2. (a) If we divide the interval [, 2] into n equal intervals of equal length, how long is each interval? (b) Write a sum which gives the right-endpoint approimation R n to the the area under the graph of f on [, 2]. (c) Use one of the formulae for the sums of powers of k to find a closed form epression for R n. (d) Take the it of R n as n tends to infinity to find an eact value for the area.

33 Worksheet # 24: Review for Eam III. (a) Find the linear approimation, L(), to f() = 2 at = 4. (b) Use the result of (a) to approimate. (c) Find the absolute error in the approimation of by using your calculator. 2. (a) Describe in words and diagrams how to use the first and second derivative tests to identify and classify etrema of a function f(). (b) Use the first derivative test to identify and classify the etrema of the function f() = Find the absolute minimum of the function f(t) = t+ t 2 on the interval [, ]. Be sure to specify the value of t where the minimum is attained. 4. For each of the following functions (i) Find the intervals on which f is increasing or decreasing. (ii) Find the local maimum and minimum values of f. (iii) Find the intervals of concavity and the inflection points. (a) f() = (b) f() = e 2 + e 5. For what values of c does the polynomial p() = 4 +c have two inflection points? One inflection point? No inflection points? 6. (a) State the Mean Value Theorem. Use complete sentences. (b) Does there eist a function f such that f() =, f(2) = 4, and f () 2 for all? 7. (a) State L Hospital s Rule for its in indeterminate form of type /. Use complete sentences, and include all necessary assumptions. (b) Evaluate e 2 (c) Evaluate + 3 ln() (d) Evaluate e 2 (e) Evaluate A poster is to have an area of 8 cm 2 with cm margins at the bottom and sides and 2 cm margins at the top. What dimensions will give the largest printed area. Be sure to eplain how you know you have found the largest area. (a) Draw a picture and write the constraint equation. (b) Write the function you are asked to maimize or minimize and determine its domain. (c) Find the maimum or minimum of the function that you found in part (c). 9. Find a positive number such that the sum of the number and twice its reciprocal is small as possible.. Let f() = 2 3 +, = 3. Apply Newton s Method to f() and initial guess to calculate 2, 3, 4.. Find the most general anti-derivative of f() = 2 + cos(2 + ). 2. Find a function with f () = sin(2), f(π) =, and f() = Consider the region bounded by the graph of f() = /, the -ais, and the lines = and = 2. Find L 3, the left endpoint approimation of this area with 3 subdivisions. 4. We know n k= k = n(n + )/2 for n =, 2,.... Find 2 k=5 (4k + ).

34 Worksheet # 25: Definite Integrals and The Fundamental Theorem of Calculus. Suppose f() d = 2, 2 f() d = 3, g() d =, and Compute the following using the properties of definite integrals: (a) (b) (c) (d) (e) Simplify 3. Find 5 g() d [2f() 3g()] d g() d f() d + f() d f() d + g() d g() d 3 f() d where f() = 3f() d 3 { 3 if < 3 if 3. 2f() d 2 2 3f() d g() d = (a) State both parts of the Fundamental Theorem of Calculus using complete sentences. (b) Consider the function f() on [, ) defined by f() = t5 dt. Argue that f is increasing. (c) Find the derivative of the function g() = 3 t5 dt on (, ). 5. Use Part I of the Fundamental Theorem of Calculus to find the derivative of the following functions: (a) g() = (b) F () = (c) h() = (d) y() = (e) G() = ( 2 + t 4 ) 5 dt cos ( t 5) dt 3 + r3 dr sin 3 (t) dt t sin(t) dt 6. Use Part II of the Fundamental Theorem of Calculus to evaluate the following integrals or eplain why the theorem does not apply: (a) d

35 (b) (c) (d) (e) 7 2 π 4 π 6 π 3 5 d e u+ du sec 2 (t) dt sin(2) sin() d 7. Below is pictured the graph of the function f(), its derivative f (), and an antiderivative f() d. Identify f(), f () and f() d. 8. Evaluate the following its by first recognizing the sum as a Riemann sum: (a) (b) (c) n i 3 n n 4 i= n n i= n n i= 3 + i n n 2 (2 + 2i n )2 n

36 Worksheet # 26: Net Change and The Substitution Method. A population of rabbits at time t increases at a rate of 4 2t + 3t 2 rabbits per year. Find the population after 8 years if there are rabbits at t =. 2. Suppose the velocity of a particle traveling along the -ais is given by v(t) = 3t 2 + 8t + 5 m/s and the particle is initially located 5 meters left of the origin. How far does the particle travel from t = 2 seconds to t = 3 seconds? After 3 seconds, where is the particle with respect to the origin? 3. Suppose an object traveling in a straight line has a velocity function given by v(t) = t 2 8t+5 km/hr. Find the displacement and distance traveled by the object from t = 2 to t = 4 hours. 4. An oil storage tank ruptures and oil leaks from the tank at a rate of r(t) = e.t liters per minute. How much oil leaks out during the first hour? 5. Evaluate the following indefinite integrals, and indicate any substitutions that you use: 4 (a) ( + 2) 3 d (d) sec 3 () tan() d (b) d (e) e sin (e ) d (c) cos 4 (θ) sin(θ) dθ (f) d 6. Evaluate the following definite integrals, and indicate any substitutions that you use: (a) (b) (c) 7 π d cos() cos (sin()) d d (d) (e) e 4 e 2 d ln e 2 d 7. Assume f is a continuous function. (a) If (b) If 9 u f() d = 4, find 3 f( 2 ) d. f() d = + e u2 for all real numbers u, find 2 f(2) d.

37 Worksheet # 27: Transcendental Functions and Other Integrals. Evaluate the following indefinite integrals: d dv (a) (d) v v 2 d (b) (e) e d 2 dt (c) + t 2 (f) 2e 2 d 2. Use the equation b = e ln(b) to find the indefinite integral b d 3. Find b so that b d is equal to (a) ln(3) (b) 3 4. Find b such that b d + 2 = π Which integral should be evaluated using substitution? Evaluate both integrals: 9 d d (a) + 2 (b) Find a relation between and u that yields = 4 + u Evaluate the following indefinite integrals, and indicate any substitutions that you use: d ln(arccos()) d (a) 2 (e) + 3 arccos() 2 cos(ln(t)) dt dt (b) (f) t t 2t 2 3 d d (c) (g) 7 2 (4 ) ln(8 2) dt (d) 4t 2 (h) e 9 2 d Evaluate the following definite integrals, and indicate any substitutions that you use: (a) (b) (c) tan(.5) tan(.5) e e 2 /5 /5 dt t d 2 + d (d) (e) (f) 3 4 /2 d arctan()( + 2 ) dt 4t /(2 2) d 6 2

38 Worksheet #28: Eponential Growth and Decay, and Area Between Curves. Solve the following equations for α: (a) 5 = e 2α (b) 4 = αe k, where k = ln(2) 7. (c), = 4, e.6α. (d) α = 2, e 36k, where k = ln(.5) The mass of substance X decays eponentially. Let m(t) denote the mass of substance X at time t where t is measured in hours. If we know m() = grams and m() = 5 grams, find an epression for the mass at time t. 3. A certain cell culture grows at a rate proportional to the number of cells present. If the culture contains 5 cells initially and 8 after 24 hours, how many cells will there be after a further 2 hours? 4. Suppose that the rate of change of the mosquito population in a pond is directly proportional to the number of mosquitoes in the pond. dp dt = KP where P = P (t) is the number of mosquitoes at time t, t is measured in days and the constant of proportionality K =.7 (a) Give the units of K. (b) If the population of mosquitoes at time t = is P () = 2. How many mosquitoes will there be after 9 days? 5. Find the area of the region between the graphs of y = 2 and y = Find the area of the regions enclosed by the graphs of y = and y = in two ways. ) By writing an integral in. 2) Solve each equation to epress in terms of y and write an integral with respect to y. 7. Find the area of the region enclosed by the graphs of y = + and y = ( 8. (a) Show that ln ( + hr) t/h) = rt. (Hint: Use L Hospital). h ( (b) From (a) show that ( + hr) t/h ) = e rt. (Hint: Use your previous result and properties of h logarithms.) (c) The compound interest formula is ( P (t) = A + r ) nt n where P (t) is the value at time t, A is the initial investment, r is the interest rate, t is the time in years, and n is the number of times compounded per year. Verify that ( ( + hr) t/h ) ) nt = n h (Hint: Use the change of variable h = n ) ( + r n Use part (b) to derive a formula involving e for continuously compounded interest. (Hint: How can you represent continuously compounded interest as a it where the interval of time between each compounding decreases to zero and the number of compoundings increases to infinity?)