Test one Review Cal 2

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Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single quantity. ln x ln x 2 ˆ. Find the following limit if it exists: lim Use when appropriate. x 5 ln( x 5).. Find the following limit if it exists: lim ln( x ). x Use when appropriate. 5. Find an equation of the tangent line to the graph of y ln x ˆ at the point (,0). 6. Differentiate the function fx ( ) ln( x 2). 7. Differentiate the function f(x) ln 8x 2 ˆ x. 8. Find the derivative of the function y ln x 2 7. 9. Find the derivative of the function x ˆ fx ( ) ln x 2. 0. Find the derivative of the function y ln ln x 2 ˆ.. Use implicit differentiation to find dy at the point dx 7, ˆ. xy 5lny 28 2. Find all points of inflection on the graph of fx ( ) ( 5x 7)ln x,x 0.. Use logarithmic differentiation to find dy dx. y 5x ( x 2). Find the indefinite integral of 5. Find the indefinite integral. x x 2 9 dx x 5 dx. x 2 x 0 6. Find dx. x 2 7. Find tan6 d. 8. Find the indefinite integral. csc( x) dx.

Name: ID: A 9. Find the indefinite integral. 20. Find cos( 2 ) sin( 2 ) d x x dx. 2. Find F( x) given F( x) 6x 7 t dt. 22. Use the Horizontal Line Test to determine whether the following statement is true or false. The function fx ( ) x is one-to-one on its 9 entire domain and therefore has an inverse function. 2. Find f ( x) if fx ( ) 2x 0. 2. Find f ( x) if fx ( ) x. 25. Find f ( x) if fx ( ) 6x 2, x 0. 26. Find dy dx x y 6y 2 8. at the point, ˆ for the equation 27. Use the functions fx ( ) x 2andgx ( ) x 7 to find the function g û f ˆ ( x). 28. Use the functions fx ( ) x 2andgx ( ) x to find the function f û g ˆ ( x). 29. Solve the following equation for x.. Find the derivative of the function fx ( ) x 6 e x. ˆ 2. Differentiate the function fx ( ) ln e5x e 2x.. Find the indefinite integral. 2xe x 2 dx. Find the indefinite integral. cosxe sin x dx 5. Evaluate the following expression without using a calculator. ˆ log 6 6. Solve the following equation for x. log x log ( x 2) 7. Use logarithmic differentiation to find dy dx. y x 8x 8. Find the indefinite integral. 6 5x dx 9. Evaluate the expression arcsin ˆ without using a 2 calculator. ln x e ( ) 0. Differentiate the function fx ( ) 5e x2. 2

Name: ID: A 0. Evaluate the expression arccos using a calculator. 2 2 ˆ without 50. Find the derivative of the function y ln cosh ˆ ( 0x). 5. Find the indefinite integral.. Evaluate the expression cos arcsin ˆ 5 without using a calculator. x 5 csch 2 x 6 6 ˆ dx 2. Write the following expression in algebraic form. cos arcsin 2x 2 ˆ ˆ. Write the following expression in algebraic form. sin arccos( 2x) ˆ. Find the indefinite integral. 9 6x 2 dx 5. Find the integral t t dt. 8 6. Evaluate the integral /6 0 6 9x 2 dx. 7. Evaluate the integral 2 9 x 2 dx. 8. Find the integral x 5 dx. x 2 9. Evaluate sinh ln ˆ ˆ 8 and cosh ln ˆ ˆ 8 in that order.

Test one Review Cal 2 Answer Section SHORT ANSWER. ANS: x ln x 2 ˆ 6 2 ˆ PTS: DIF: Medium REF: 5..5 OBJ: Write a logarithmic expression as a single quantity NOT: Section 5. 2. ANS: ˆ x ln x 2 ˆ PTS: DIF: Medium REF: 5..5 OBJ: Write a logarithmic expression as a single quantity NOT: Section 5.. ANS: does not exist PTS: DIF: Medium REF: 5..9 OBJ: Evaluate limits involving logarithmic functions NOT: Section 5.. ANS: PTS: DIF: Medium REF: 5..9 OBJ: Evaluate limits involving logarithmic functions NOT: Section 5. 5. ANS: y ( x ) PTS: DIF: Medium REF: 5.. OBJ: Write an equation of a line tangent to the graph of a function at a specified point NOT: Section 5.

6. ANS: x 2 PTS: DIF: Medium REF: 5..8 OBJ: Differentiate a logarithmic function using the chain rule NOT: Section 5. 7. ANS: 6x 8x 2 x PTS: DIF: Medium REF: 5..50 OBJ: Differentiate a logarithmic function using the chain rule NOT: Section 5. 8. ANS: dy dx x x 2 7 PTS: DIF: Medium REF: 5..5 OBJ: Differentiate a logarithmic function using the chain rule NOT: Section 5. 9. ANS: f ( x) x 2x x 2 PTS: DIF: Medium REF: 5..57 OBJ: Differentiate a logarithmic function using the chain rule and quotient rule NOT: Section 5. 0. ANS: dy dx 2 x ln x 2 ˆ PTS: DIF: Medium REF: 5..6 OBJ: Differentiate a logarithmic function using the chain rule NOT: Section 5.. ANS: PTS: DIF: Medium REF: 5..88 OBJ: Differentiate an equation using implicit differentiation and evaluate at a specified point NOT: Section 5. 2

2. ANS: 7 5,ln 7 ˆ 5 PTS: DIF: Medium REF: 5..9 OBJ: Identify all points of inflection for a function NOT: Section 5.. ANS: 0x 7 ( x 2) PTS: DIF: Medium REF: 5..06 OBJ: Differentiate a function using logarithmic differentiation NOT: Section 5.. ANS: lnx 5 C PTS: DIF: Easy REF: 5.2. NOT: Section 5.2 5. ANS: 8 ln x2 9 C PTS: DIF: Medium REF: 5.2.7 NOT: Section 5.2 6. ANS: x 2 x 0 dx x 2 2 x2 26x 22ln x 2 C PTS: DIF: Difficult REF: 5.2.5 NOT: Section 5.2 7. ANS: tan6 d 6 ln cos 6 C PTS: DIF: Easy REF: 5.2.2 NOT: Section 5.2

8. ANS: lncsc ( x) cot x ( ) C PTS: DIF: Medium REF: 5.2. NOT: Section 5.2 9. ANS: lnsin 2 2 ( ) C PTS: DIF: Medium REF: 5.2.7 NOT: Section 5.2 20. ANS: ˆ x x dx x 2 2ln x 2 2 x C PTS: DIF: Medium REF: 5.2.6 OBJ: Evaluate the definite integral of a function using a computer algebra system NOT: Section 5.2 2. ANS: F( x) 7 x PTS: DIF: Easy REF: 5.2.69 OBJ: Calculate the derivative of an integral function NOT: Section 5.2 22. ANS: true PTS: DIF: Medium REF: 5.. OBJ: Recognize invertible functions MSC: Application NOT: Section 5. 2. ANS: f ( x) 2 x 5 6 PTS: DIF: Easy REF: 5.2.2a OBJ: Construct the inverse of a function NOT: Section 5. 2. ANS: f ( x) ( x ) PTS: DIF: Medium REF: 5..26a OBJ: Construct the inverse of a function NOT: Section 5.

25. ANS: f ( x) x 6 PTS: DIF: Medium REF: 5..28a OBJ: Construct the inverse of a function NOT: Section 5. 26. ANS: 9 PTS: DIF: Medium REF: 5..85 OBJ: Evaluate a derivative at a point using the inverse function NOT: Section 5. 27. ANS: x 5 PTS: DIF: Easy REF: 5..9 OBJ: Construct the inverse of a composition of functions NOT: Section 5. 28. ANS: x PTS: DIF: Easy REF: 5..9 OBJ: Construct the inverse of a composition of functions NOT: Section 5. 29. ANS: x PTS: DIF: Easy REF: 5..2 OBJ: Solve an exponential equation NOT: Section 5. 0. ANS: 0xe x2 PTS: DIF: Medium REF: 5..2 OBJ: Differentiate an exponential function using the chain rule NOT: Section 5.. ANS: x 5 e x ( x 6) PTS: DIF: Medium REF: 5..7 OBJ: Differentiate an exponential function using the chain rule and product rule NOT: Section 5. 5

2. ANS: 5e 5x e 5x 2e2x e 2x PTS: DIF: Medium REF: 5..52 OBJ: Differentiate a logarithmic function using the chain rule NOT: Section 5.. ANS: 2 ex C PTS: DIF: Medium REF: 5..0 OBJ: Evaluate the indefinite integral of an exponential function using substitution NOT: Section 5.. ANS: esin x C PTS: DIF: Medium REF: 5..25 OBJ: Evaluate the indefinite integral of an exponential function using substitution NOT: Section 5. 5. ANS: PTS: DIF: Medium REF: 5.5. OBJ: Evaluate a logarithmic expression NOT: Section 5.5 6. ANS:, PTS: DIF: Medium REF: 5.5.2b OBJ: Solve a logarithmic equation NOT: Section 5.5 7. ANS: 8x 8x ( ln x ) PTS: DIF: Medium REF: 5.5.68 OBJ: Differentiate a function using logarithmic differentiation NOT: Section 5.5 8. ANS: 5ln6 65x C PTS: DIF: Medium REF: 5.5.75 OBJ: Evaluate the indefinite integral of an exponential function using substitution NOT: Section 5.5 6

9. ANS: 6 PTS: DIF: Easy REF: 5.6.5 OBJ: Evaluate an inverse trigonometric expression NOT: Section 5.6 0. ANS: PTS: DIF: Easy REF: 5.6.9 OBJ: Evaluate an inverse trigonometric expression NOT: Section 5.6. ANS: 5 PTS: DIF: Medium REF: 5.6.8b OBJ: Evaluate an expression involving an inverse trigonometric expression NOT: Section 5.6 2. ANS: x PTS: DIF: Medium REF: 5.6.27 OBJ: Convert an inverse trigonometric expression to an algebraic expression NOT: Section 5.6. ANS: x 2 PTS: DIF: Medium REF: 5.6.27 OBJ: Convert an inverse trigonometric expression to an algebraic expression NOT: Section 5.6. ANS: x ˆ arcsin C PTS: DIF: Medium REF: 5.7. OBJ: Evaluate the indefinite integral involving an inverse trigonometric function NOT: Section 5.7 5. ANS: 8 arctan t2 9 dt C PTS: DIF: Medium REF: 5.7. OBJ: Evaluate the indefinite integral involving an inverse trigonometric function NOT: Section 5.7 7

6. ANS: 8 9 PTS: DIF: Easy REF: 5.7.25 OBJ: Evaluate a definite integral involving inverse trigonometric functions NOT: Section 5.7 7. ANS: PTS: DIF: Easy REF: 5.7.28 OBJ: Evaluate a definite integral involving inverse trigonometric functions NOT: Section 5.7 8. ANS: 2 ln x 2 ˆ 5arctan( x) C PTS: DIF: Difficult REF: 5.7.2 OBJ: Evaluate the indefinite integral involving an inverse trigonometric function and a logarithmic function NOT: Section 5.7 9. ANS: 6 6, 65 6 PTS: DIF: Medium REF: 5.8.a OBJ: Evaluate a hyperbolic function NOT: Section 5.8 50. ANS: dy dx 0tanh ( 0x ) PTS: DIF: Medium REF: 5.8.2 OBJ: Differentiate a hyperbolic function NOT: Section 5.8 5. ANS: ˆ coth x6 6 C PTS: DIF: Medium REF: 5.8.5 OBJ: Evaluate an indefinite integral involving hyperbolic functions NOT: Section 5.8 8

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