(A) when x = 0 (B) where the tangent line is horizontal (C) when f '(x) = 0 (D) when there is a sharp corner on the graph (E) None of the above

Transcription

1 AP Physics C - Problem Drill 10: Differentiability and Rules of Differentiation Question No. 1 of 10 Question 1. A derivative does not eist Question #01 (A) when 0 (B) where the tangent line is horizontal (C) when f '() 0 (D) when there is a sharp corner on the graph (E) None of the above The derivative may eist when 0 depending on the function. When the tangent line is horizontal, the derivative equals zero. The derivative equaling zero does not imply that it does not eist. D. Correct! The derivative does not eist at sharp corners on the graph. One of the given answer choices is correct. The derivative does not eist at sharp corners on the graph.

2 Question No. of 10 Question. Find the derivative of f () 16. Question #0 (A) 0 (B) 1 (C) 16 (D) -16 (E) None of the above A. Correct! The derivative of a constant is zero. The function is a constant function. The function is a constant function. The function is a constant function. One of the given answer choices is correct. The function f () 16 is a constant function. The derivative of a constant is 0.

3 Question No. 3 of 10 Question 3. Find the derivative of f () -41. Question #03 (A) -41 (B) 0 (C) -41 (D) 41 A. Correct! The derivative of k is k. The derivative of a constant times is just the constant. The derivative of a constant times is just the constant. The derivative of a constant times is just the constant. One of the given answer choices is correct. Using the rule: If f () k, then f '() k, the derivative of f () -41 is f '() -41.

4 Question No. 4 of 10 Question 4. Find the derivative of f () 7. Question #04 (A) 7 (B) 7 7 (C) 6 6 (D) 7 6 Use the power rule for derivatives. Use the power rule for derivatives. Use the power rule for derivatives. D. Correct! This is the derivative using the power rule. One of the given answer choices is correct. n Using the rule: If f( ), then 6 f '( ) 7. n 1 f '( ) n, the function ( ) 7 f has a derivative of

5 Question No. 5 of 10 Question 5. Find the derivative of f( ) 1. Question #05 (A) f '( ) 1 1 (B) f '( ) 3 1 (C) f '( ) 1 (D) f '( ) 3 Rewrite the function using eponents and then take the derivative. B. Correct! By rewriting the function using eponents, the derivative can be found. Rewrite the function using eponents and then take the derivative. Rewrite the function using eponents and then take the derivative. One of the given answer choices is correct. We first rewrite 1 as 1/. Now, using the power rule, 1 3/ 1 f '( ). 3

6 Question No. 6 of 10 Question 6. Find the derivative of f( ) e Question #06 (A) f '( ) e (B) 1 f '( ) e (C) f '( ) e (D) f '( ) e The derivative of the second term is incorrect.! The derivative of the first term is incorrect. C. Correct! Take the derivative on each term and sum them up. The derivative of the second term is incorrect. One of the given answer choices is correct. Please try again. The derivative of e is e and the derivative of is. So the overall derivative is f '( ) e

7 Question No. 7 of 10 Question 7. Find the derivative of f( ) sin( ) cos( ). Question #07 (A) f '( ) cos( ) sin( ) (B) f '( ) cos( ) + sin( ) (C) f '( ) sin( ) cos( ) (D) 1 The derivative of cos() is -sin(). B. Correct! This is the correct derivative. This the same as the original function. Use the formulas for the derivative of sin() and cos() One of the given answers is correct. Please try again. Recall that (sin( ))' cos( ) and ( ) f '( ) cos( ) ( sin( )) cos( ) + sin( ). cos( ) ' sin( ). The derivative of f( ) sin( ) cos( ) is

8 Question No. 8 of 10 Question 8. Find the derivative of sin( f( ) tan( ) ) cos( ) Question #08 (A) sin( ) (B) cos( ) (C) sec ( ) cos( ) (D) sin( ) Use the quotient rule. Use the quotient rule. C. Correct! By using the quotient rule, this is the correct derivative. Use the quotient rule. One of the given answers is correct. The derivative is: sin( ) f '( ) cos( ) (sin )' cos sin (cos )' cos cos cos sin ( sin ) cos cos + sin cos 1 cos sec

9 Question No. 9 of 10 Question 9. Find the derivative of f( ) e Question #09 (A) e (B) + e (C) e (D) e + e Use the product rule. Use the product rule. Use the product rule. D. Correct! By using the product rule, this is the correct derivative. One of the given answers is correct. The derivative is: f '( ) ( e )' ( )' e + ( e )' e + e

10 Question No. 10 of 10 5 Question 10. Find the derivative of f( ) 4. Question #10 (A) 4 15 (B) (C) 3 0 (D) A. Correct! By simplifying the function, this is the correct derivative. Simplify the function and then take the derivative. Simplify the function and then take the derivative. Simplify the function and then take the derivative. One of the given answers is correct. 5 To find the derivative of f( ) 4, first simplify to get 5 3 f( ) 5. The derivative is 3 4 f '( ) 15.