Study Guide and Review - Chapter 12

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Choose the correct term to complete each sentence. 1. The slope of a nonlinear graph at a specific point is the and can be represented by the slope of the tangent line to the graph at that point.

The process of evaluating an integral is called. The process of evaluating an integral is called integration. 3.

1 Choose the correct term to complete each sentence. 1. The slope of a nonlinear graph at a specific point is the and can be represented by the slope of the tangent line to the graph at that point. The slope of a nonlinear graph at a specific point is the instantaneous rate of change and can be represented by the slope of the tangent line to the graph at that point. 2. The process of evaluating an integral is called. The process of evaluating an integral is called integration. 3. Limits of polynomial and rational functions can be found by, so long as the denominator of the rational function evaluated at c is not 0. Limits of polynomial and rational functions can be found by direct substitution, so long as the denominator of the rational function evaluated at c is not The function F(x) is the of f (x). The function F(x) is the antiderivative of f (x). 5. Since it is not possible to determine the limit of the function with 0 in the denominator, it is customary to describe the resulting fraction as having a(n). Since it is not possible to determine the limit of the function with 0 in the denominator, it is customary to describe the resulting fraction as an indeterminate form. 6. To find the limits of rational functions at infinity, divide the numerator and denominator by the power of x that occurs in the function. To find the limits of rational functions at infinity, divide the numerator and denominator by the highest power of x that occurs in the function. 7. The process of finding a derivative is called. The process of finding a derivative is called differentiation. esolutions Manual - Powered by Cognero Page 1

2 8. If a function is preceded by a(n), then you are to take the derivative of the function. If a function is preceded by a(n) differential operator, then you are to take the derivative of the function. 9. The velocity or speed achieved at any specific point in time is the. The velocity or speed achieved at any specific point in time is the instantaneous velocity. 10. The indefinite integral of f (x) is defined by =. The indefinite integral of f (x) is defined by = F(x) + C. Estimate each limit using a graph. Support your conjecture using a table of values. 11. Graph f (x) =. The graph of f (x) = is 1. suggests that as x approaches 3, f (x) approaches 1. Therefore, we can estimate that Support Numerically Make a table of values for f, choosing x-values that approach 3 by using some values slightly less than 3 and some values slightly greater than 3. The pattern of outputs suggests that as x approaches 3 from the left or from the right, f (x) approaches 1. This supports our graphical analysis. esolutions Manual - Powered by Cognero Page 2

3 12. Analyze Graphically Graph f (x) =. The graph of f (x) = estimate that suggests that as x approaches 1, f (x) approaches 1.5. Therefore, we can is 1.5. Support Numerically Make a table of values for f, choosing x-values that approach 1 by using some values slightly less than 1 and some values slightly greater than 1. The pattern of outputs suggests that as x approaches 1 from the left or from the right, f (x) approaches 1.5. This supports our graphical analysis. Estimate each one-sided or two-sided limit, if it exists. 13. Graph f (x) = The graph of f (x) = suggests that = 5. esolutions Manual - Powered by Cognero Page 3

4 14. Graph f (x) = The graph of f (x) = suggests that = and =. Since the left- and right-hand limits of f (x) as x approaches 4 approach different values, the exist. does not Estimate each limit, if it exists. 15. Graph f (x) = The graph of f (x) = suggests that =. esolutions Manual - Powered by Cognero Page 4

5 16. Graph f (x) = The graph of f (x) = suggests that does not exist Use the properties of limits to evaluate each limit esolutions Manual - Powered by Cognero Page 5

6 19. Use direct substitution, if possible, to evaluate each limit. If not possible, explain why not. This is the limit of a rational function. Since the denominator of this function is 0 when x = 25, the limit cannot be found by direct substitution Evaluate each limit. 22. esolutions Manual - Powered by Cognero Page 6

7 Find the slope of the lines tangent to the graph of each function at the given points. 23. y = 6 x; ( 1, 7) and (3, 3) Find the slope of the line tangent to the graph at ( 1, 7). Find the slope of the line tangent to the graph at (3, 3). esolutions Manual - Powered by Cognero Page 7

8 24. y = x 2 + 2; (0, 2) and ( 1, 3) Find the slope of the line tangent to the graph at (0, 2). Find the slope of the line tangent to the graph at ( 1, 3). esolutions Manual - Powered by Cognero Page 8

9 The distance d an object is above the ground t seconds after it is dropped is given by d(t). Find the instantaneous velocity of the object at the given value for t. 25. y = x 2 + 3x An equation for the slope of the graph at any point is m = 2x y = x 3 + 4x An equation for the slope of the graph at any point is m = 3x esolutions Manual - Powered by Cognero Page 9

10 Find the instantaneous velocity if the position of an object in feet is defined as h(t) for given values of time t given in seconds. 27. h(t) = 15t + 16t 2 ; t = 0.5 To find the instantaneous velocity, let t = 0.5 and apply the formula for instantaneous velocity. The instantaneous velocity of the object is 31 feet per second. 28. h(t) = 16t 2 35t + 400; t = 3.5 To find the instantaneous velocity, let t = 3.5 and apply the formula for instantaneous velocity. The instantaneous velocity of the object is 147 feet per second. esolutions Manual - Powered by Cognero Page 10

11 Find an equation for the instantaneous velocity v(t) if the path of an object is defined as h(t) for any point in time t. 29. h(t) = 12t 2 5 Apply the formula for instantaneous velocity. The instantaneous velocity of the object at time t is v(t) = 24t. 30. h(t) = 8 2t 2 + 3t Apply the formula for instantaneous velocity. The instantaneous velocity of the object at time t is v(t) = 4t + 3. esolutions Manual - Powered by Cognero Page 11

12 Evaluate limits to find the derivative of each function. Then evaluate the derivative of each function for the given values of each variable. 31. g(t) = t 2 + 5t + 11; t = 4 and 1 The derivative of g(x) is g (t) = 2t + 5. Evaluate g (t) for t = 4 and m(j ) = 10j 3; j = 5 and 3 The derivative of m(j ) is m (j ) = 10. Evaluate m (j ) for j = 5 and 3. The derivative of m(j ) is constant. Therefore, it will be 10 at any given value. esolutions Manual - Powered by Cognero Page 12

13 Find the derivative of each function. 33. p (v) = 9v z(n) = 4n 2 + 9n esolutions Manual - Powered by Cognero Page 13

14 Use the quotient rule to find the derivative of each function. 37. f(m) = Let g(m) = 5 3m and h(m) = 5 + 2m. So,. Use g(m), g (m), h(m), and h (m) to find the derivative of f (m). esolutions Manual - Powered by Cognero Page 14

15 38. Let a(q) = 2q 4 q and b(q) = q So,. Use a(q), a (q), b(q), and b (q) to find the derivative of m(q). esolutions Manual - Powered by Cognero Page 15

1-1 Functions < x 64 SOLUTION: 9. { 0.25, 0, 0.25, 0.50, } SOLUTION: 12. all multiples of 8 SOLUTION: SOLUTION:

1-1 Functions < x 64 SOLUTION: 9. { 0.25, 0, 0.25, 0.50, } SOLUTION: 12. all multiples of 8 SOLUTION: SOLUTION: Write each set of numbers in set-builder and interval notation, if possible. 3. x 4 The set includes all real numbers less than or equal to 4. In set-builder notation this set can be described as {x x