The italicized phrase before each sample problem indicates a similar problem in James Stewart. 2007. Essential Calculus: Early Transcendentals. Thomson Brooks/Cole: Belmont, CA.
Similar to 2.1.6:
2.1 Obtain the equation for the line tangent at the given point of the given function.
f(x) = 7x / (2x + 3)^3 at (2, 2/49)
Similar to 2.1.38:
2.2 The number of records R sold by a rising band by year is given in thousands of records below.
Year 2006 2007 2008 2009 2010
R 754 1006 1099 2097 2201
a. What was the average rate of growth in sales for these time periods?
| (i) from 2008 to 2010 | (ii) from 2007 to 2008 |
| (iii) from 2006 to 2008 | (iv) from 2008 to 2009 |
What are the units for these answers?
b. Average two average rates to estimate the instantaneous growth rate for 2008. What are its units?
c. Plot the points in the table and sketch the best (smoothest) curve you can through them. Then measure the slope of a tangent line to obtain another estimate of the instantaneous growth rate for 2008.
Similar to 2.2.24abc:
2.3 Transformations
a. Graph f(x) = sqrt (5 - 2x) in three transformations starting with the graph of
g(x) = sqrt (x).
b. Sketch f'(x) using the graph of f(x).
c. Obtain f'(x) using the definition of a derivative. Give the domains for f(x) and f'(x).
Similar to 2.2.34:
2.4 Below are the graphs of f, f', f", and f"'. Identify each and justify your choices.
Similar to 2.3.60:
2.5 Lines Tangent to a Curve
a. Obtain the equation for each line through the point (2,-1) that's tangent to
y = x^2 - 2x.
b. Obtain the equation for each line through the point (1,3) that's tangent to
y = x^2 - 2x.
c. Drawing and referring to a graph, succinctly but thoroughly explain why the number of equations obtained in step b differ from that obtained in step a.
Similar to 2.4.46:
2.6 Assume f(x) is differentiable and find the derivative for each of the functions below in terms of f'(x), f(x), and x.
| a. y = (5x^3) f(x) | b. y = f(x) / (7x^4) |
| c. y = 3x^5 / f(x) | d. y = [9 + 2x f(x)] / cuberoot (x) |
Similar to 2.5.52:
2.7 Let h(x) = f(2f(x)) and g(x) = f(3x^2). Use the graph of f below to estimate these derivatives.
| (a) h'(2.5) | (b) g'(2.5) |
Similar to 2.5.56:
2.8 Higher order derivatives
Let g be an arbitrary, twice differentiable function and f(x) = 3x^4 g(2x^3). Obtain f''(x) in terms of x, g, g', and g''.
Similar to 2.6.38:
2.9 Tangents to a curve
Prove that, for every line tangent to 3 sqrt (2x) + 3 sqrt (2y) = 23, adding the x- and y-intercepts yields 529. Can you explain how this problem is similar to 2.6.38?
Similar to 2.8.22:
2.10 Measurement error effects on calculated quantities
A square is cut from a tin sheet with sides 2 1/2 meters each measured accurate to within half a centimeter.
a. Determine the maximum error, using differentials, in the calculated area of the square.
b. Determine the relative and percent errors.
c. Give two assumptions made in parts a and b.
