- 3.1: State each of the following differentiation rules both in symbols a...
- 3.2: State the derivative of each function. (a) (b) (c) (d) (e) (f) (g) ...
- 3.3: (a) How is the number defined? (b) Express as a limit. (c) Why is t...
- 3.4: (a) Explain how implicit differentiation works. (b) Explain how log...
- 3.5: What are the second and third derivatives of a function f ? If f is...
- 3.6: (a) Write an expression for the linearization of at a. (b) If , wri...
- 3.7: Determine whether the statement is true or false. If it is true, ex...
- 3.8: Determine whether the statement is true or false. If it is true, ex...
- 3.9: Determine whether the statement is true or false. If it is true, ex...
- 3.10: Determine whether the statement is true or false. If it is true, ex...
- 3.11: Determine whether the statement is true or false. If it is true, ex...
- 3.12: Determine whether the statement is true or false. If it is true, ex...
- 3.13: Determine whether the statement is true or false. If it is true, ex...
- 3.14: Calculate .y 1 sinx sin x y
- 3.15: Calculate .xy4 + 4 x 2 y x 3y
- 3.16: Calculate .y lncsc 5x 4
- 3.17: Calculate .y c sec 2 1 tan 2
- 3.18: Calculate .x 2 y cos y sin 2y xy s
- 3.19: Calculate .y e cxc sin x cos x x 2
- 3.20: Calculate .y lnx 2 ex y e cxc
- 3.21: Calculate .y e
- 3.22: Calculate .y sec1 x 2 y
- 3.23: Calculate .y 1 x x sx 1 1 y sec
- 3.24: Calculate .y 1s 3 y 1 x x sx 1
- 3.25: Calculate .sinxy x = 2 y
- 3.26: Calculate .y ssin sx 2
- 3.27: Calculate .y log51 2x si
- 3.28: Calculate .y cos x x y
- 3.29: Calculate .y ln sin x 1 2 sin2 x y
- 3.30: Calculate .y x 2 1 4 2x 1 3 3x 1 5 y ln si
- 3.31: Calculate .y x tan 1 4x y
- 3.32: Calculate .y ecos x cosex y
- 3.33: Calculate .y ln sec 5x tan 5x
- 3.34: Calculate .y 10tan
- 3.35: Calculate .y cot3x 2 5 y
- 3.36: Calculate .y st lnt 4 y
- 3.37: Calculate .y sin(tan s1 x y
- 3.38: Calculate .y arctan(arcsin sx ) 3
- 3.39: Calculate .y tan y y 1 2 sin y
- 3.40: Calculate .xe = y y 1 2
- 3.41: Calculate .sx 1 2 x 5 x 3 7 xe y
- 3.42: Calculate .y x 4 x 4 4 y
- 3.43: Calculate .y x sinhx 2 y
- 3.44: Calculate .y sin mx x y
- 3.45: Calculate .
- 3.46: Calculate .
- 3.47: Calculate .
- 3.48: Calculate .
- 3.49: If , find .
- 3.50: If , find .
- 3.51: Find if .
- 3.52: Find if .
- 3.53: Use mathematical induction to show that if , then
- 3.54: Evaluate limt l 0t3tan3 2tf
- 3.55: Find an equation of the tangent to the curve at the given point.
- 3.56: Find an equation of the tangent to the curve at the given point.
- 3.57: Find an equation of the tangent to the curve at the given point.
- 3.58: Find an equation of the tangent to the curve at the given point.
- 3.59: Find an equation of the tangent to the curve at the given point.
- 3.60: If , find . Graph and on the same screen and comment.
- 3.61: (a) If , find . (b) Find equations of the tangent lines to the curv...
- 3.62: (a) If , , find and . ; (b) Check to see that your answers to part ...
- 3.63: At what points on the curve , , is the tangent line horizontal?
- 3.64: Find the points on the ellipse where the tangent line has slope 1.
- 3.65: If , show that .
- 3.66: (a) By differentiating the double-angle formula obtain the double-a...
- 3.67: Suppose that and , where , , , , and . Find (a) and (b) .
- 3.68: If and are the functions whose graphs are shown, let , , and . Find...
- 3.69: Find f' in terms of g' .
- 3.70: Find f' in terms of g' .
- 3.71: Find f' in terms of g' .
- 3.72: Find f' in terms of g' .
- 3.73: Find f' in terms of g' .
- 3.74: Find f' in terms of g' .
- 3.75: Find f' in terms of g' .
- 3.76: Find f' in terms of g' .
- 3.77: Find f' in terms of g' .
- 3.78: Find f' in terms of g' .
- 3.79: Find f' in terms of g' .
- 3.80: (a) Graph the function in the viewing rectangle by . (b) On which i...
- 3.81: At what point on the curve is the tangent horizontal?
- 3.82: (a) Find an equation of the tangent to the curve that is parallel t...
- 3.83: Find a parabola that passes through the point and whose tangent lin...
- 3.84: The function , where a, b, and K are positive constants and , is us...
- 3.85: An equation of motion of the form represents damped oscillation of ...
- 3.86: A particle moves along a horizontal line so that its coordinate at ...
- 3.87: A particle moves on a vertical line so that its coordinate at time ...
- 3.88: The volume of a right circular cone is , where is the radius of the...
- 3.89: The mass of part of a wire is kilograms, where is measured in meter...
- 3.90: The cost, in dollars, of producing units of a certain commodity is ...
- 3.91: The volume of a cube is increasing at a rate of 10 . How fast is th...
- 3.92: A paper cup has the shape of a cone with height 10 cm and radius 3 ...
- 3.93: A balloon is rising at a constant speed of . A boy is cycling along...
- 3.94: A waterskier skis over the ramp shown in the figure at a speed of ....
- 3.95: The angle of elevation of the Sun is decreasing at a rate of . How ...
- 3.96: (a) Find the linear approximation to near 3. (b) Illustrate part (a...
- 3.97: (a) Find the linearization of at . State the corresponding linear a...
- 3.98: Evaluate if , , and .
- 3.99: A window has the shape of a square surmounted by a semicircle. The ...
- 3.100: Express the limit as a derivative and evaluate.
- 3.101: Express the limit as a derivative and evaluate.
- 3.102: Express the limit as a derivative and evaluate.
- 3.103: Evaluate limx l 0s1 tan x s1 sin xx 3.
- 3.104: Suppose is a differentiable function such that and . Show that .
- 3.105: Find if it is known that ddx f 2x x 2f
- 3.106: Show that the length of the portion of any tangent line to the astr...
Solutions for Chapter 3: Differentiation Rules
Full solutions for Calculus, | 5th Edition
The farthest point from the Sun in a planet’s orbit
The notation PQ denoting the directed line segment with initial point P and terminal point Q.
The real number multiplied by the variable(s) in a polynomial term
See Geometric sequence.
See Cartesian coordinate system.
Direction vector for a line
A vector in the direction of a line in three-dimensional space
Focal length of a parabola
The directed distance from the vertex to the focus.
Matrix, m x n
A rectangular array of m rows and n columns of real numbers
The numbers 1, 2, 3, . . . ,.
See Linear programming problem.
The number zero on a number line, or the point where the x- and y-axes cross in the Cartesian coordinate system, or the point where the x-, y-, and z-axes cross in Cartesian three-dimensional space
An arrangement of elements of a set, in which order is important.
An equation in r and ?.
Right-hand limit of ƒ at x a
The limit of ƒ as x approaches a from the right.
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the right-hand end point of each subinterval.
Solution set of an inequality
The set of all solutions of an inequality
Use a graphical method, including use of a hand sketch or use of a grapher. When appropriate, the approximate solution should be confirmed algebraically
The function y = tan x
The line segment whose endpoints are the vertices of a hyperbola.
A mean calculated in such a way that some elements of the data set have higher weights (that is, are counted more strongly in determining the mean) than others.