Assignment 1: Limit Definition of the Derivative and Power Rule
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Unit 1 Assignments
Note: An ‘*’ by the problem # means you can use your calculator. Otherwise, don’t!
Assignment 1: Evaluating Limits
1) 108, 3 2) 129, 44 3) 72, 2 4) 107, 1 5) 2, 5 6) 78, 16 7) 181, 6 8) 54, 38 9) 1,1
Assignment 2: Derivatives as Limits
(x h)3 x3 1.) The lim at the point x = 2 is: h0 h
(A) 36 (B) 12 (C) 8 (D) 2 (E) 0
Evaluate each limit.
3(x h) 2 2(x h) (3x 2 2x) 2.) lim h0 h (m b) 2 (m) 2 3.) lim b0 b (3 x)5 (3)5 4.) lim x0 x
Extra Credit: Textbook: p. 102 # 11, 15, 19
Assignment 3: The Power Rule
In 1 and 2, determine the function the limit finds the derivative for. Then, find the derivative using the power rule.
(x h) 2 2(x h) (x 2 2x) 1.) lim h0 h (3 h)2 4(3 h) (32 4 3) 2.) lim h0 h
2(x h)5 5(x h)3 2x 5 5x 3 3.) lim is: h0 h
(A) 0 (B) 10x 3 15x (C) 10x 4 15x 2 (D) 10x 4 15x 2 (E) -10x 4 15x 2
4) 152, 21
Extra Credit: Textbook: p. 113 # 6, 8, 10, 13, 14, 15, 44
Assignment 4: More on Derivatives as Slopes
In 1 and 2, find the slope of the curve at the given point. Use the Power Rule. 1.) f(x) = 3x 2 - x + 2 at (1, 4) 1 2.) f(x) = x x x at (1, 1) x
In 3 and 4, find the slope of the curve at the given x value. Use nDerive. 3.)* y = 3x sin (x 2 ) at x = 3 3 4.)* y = x at x = 4 x
5.)* The function f(x) = tan(3 x ) has one zero in the interval [0, 1.4]. The derivative at this point is
(A) 0.411 (B) 1.042 (C) 3.451 (D) 3.763 (E) undefined (HINT: Use graph-root to find the zeros, then nder to find deriv.)
In 6 and 7, find the equation of the line tangent to the given curve at the given point. 6.) y = 3x 2 + 2x + 1 at x = 0 3 7.) y = x at x = 4 x
In 8 and 9, find the equation of the line normal to the given curve at the given point. 8.) y = 6 - x 2 at x = 3 9.) y = 4 + 2x - x 3 at x = 1
| x h | | x | 10.) The lim at x = 3 is: h0 h (A) 0 (B) 1 (C) 3 (D) –1 (E) nonexistent 11.) Let f(x) be a differentiable function defined only on the interval -2 x 10 . The table below gives the value of f(x) and its derivative f’(x) at several points of the domain. X -2 0 2 4 6 8 10 F(x) 26 27 26 23 18 11 2 F’(x) 1 0 -1 -2 -3 -4 -5
The line tangent to the graph of f(x) and parallel to the segment between the endpoints intersects the y-axis at the point
(A) (0, 27) (B) (0, 28) (C) (0, 31) (D) (0, 36) (E) (0, 43) (HINT: Find the slope of the segment connecting the endpoints first)
12.) If the line y = 4x + 3 is tangent to the curve y = x 2 + c, then c is:
(A)2 (B) 4 (C) 7 (D) 11 (E) 15 (HINT: The slopes must match AND they must share a point)
13.) Let h be a function defined for all x 0 such that h(4) = -3 and the derivative of h is: h’(x) = x 2 2 for all x 0. Write an equation for the line tangent to the graph of h at x = 4. (This x question is from the 2001 exam: FR #4c.)
14.) Let f be the function given by f(x) = 3x4 + x3 –21x2. Write an equation of the line tangent to the graph of f at the point (2, -28). (This question is from the 1994 exam: FR #1a.)
15.) Let f be the function defined by f(x) = 3x5 – 5x3 + 2. Write the equation of each horizontal tangent line to the graph of f. Recall that horizontal tangent lines have a slope = 0. (This question is from the 1992 exam: FR 1c.)
Assignment 5: More on Derivatives as Slopes
In 1 and 2, determine whether the given function is differentiable for all x values. x 2 3 x 1 4 2x x 2 x 0 1.) f(x) = 2.) f(x) = 2 3x 2 x 1 x 2x 1 x 0
In 3 and 4, find the values of a and b that make the function differentiable for all values of x. x 3 1 x 0 2 x x 2 x 1 3.) f(x) = 4.) f(x)= ax b x 0 ax b x 1
In 5 and 6, find the x values for which f(x) has no derivative. 5.) f(x) = |3x + 2| + 4 6.) f(x) = |-3 – 6x| Evaluate. | x h | | x | 7.) lim [Hint: This will be a piecewise function!] h0 h
Assignment 6: The Product and Quotient Rules
1.) Find the equation of the line tangent to y = (x 2 2x 6)(x3 2x 2 x 4) at x = 1 1 x 2.) Find the equation of the line normal to y = x at x = 4. x 1
3.) Determine the values of x for which this function has no derivative.
2 3 2 (x 1)(x x ) x 1 y = 2x 2 1 x 1 x 1 x 1 x 2
kx 8 4.) The equation of the line tangent to the curve y = at x = -2 is y = x + 4. What is the value of k? k x (Remember the slopes must match and both curves must contain the same point.)
(A)–3 (B) –1 (C) 1 (D) 3 (E) 4
3x 4 5.) The equation of the line tangent to the curve y = at the point (1, 7) is 4x 3 (A) y + 25x = 32 (B) y – 31x = -24 (C) y – 7x = 0 (D) y + 5x = 12 (E) y – 25x = -18
6.)* Let f and g be functions that are differentiable for all real numbers x with f (x) g(x) = . If the equation of the line tangent to f at x = 1 is y = 2x – 3, what is the equation of x 1 the line tangent to the graph of g at x = 1? (HINT: Rewrite g(x) as f (x) and use the product x rule to find g’(1). The slope of f at x = 1 is given in the equation of the tangent line to f at x = 1.)
(A) y = 3x – 4 (B) y = x – 2 (C) y = 2x + 3 (D) y = 3x – 2 (E) y = 2x -3
7.) If f(x) = x 3 x , then f’(x) = 7 1 1 2 3 4 1 1 (A) 4x 3 (B) x 3 (C) x 3 (D) x 3 (E) x 3 7 3 3 3
1 x dy 8.) If y = , then = x 1 dx 1 2 2x (A) –1 (B) 0 (C) (D) (E) x 1 x 1 (x 1) 2
Extra Credit: Textbook: p. 124 #1-4, 7-10, 26, 28
Assignment 7: Trig Derivatives
In 1-6, find the equation of the line tangent to the curve at the x value. No calculators. Answers must be in exact form (no decimal approximations).
1.) y = sin x at x = 6 2.) y = cos x at x = 4 3.) y = tan x at x = 3 4.) y = cot x at x = 6 5.) y =sec x at x = 6 6.) y = csc x at x = 3
In 7-10, use the Product/Quotient Rules to find f’(x).
7.) f(x) = (x 2 x) sin x
csc x 8.) f(x)= 1 x
9.) f(x) = tan x( x 4 x 3 )
sec x 10.) f(x) = cot x
cot(x h) cot(x) 11.) Evaluate lim h0 h sec( h) sec( ) 12.) Evaluate lim 3 3 h0 h
13.) Find a and b values which guarantee y has a derivative at all points. x sin x cos x 3 y = ax b x 3 sin 2 x 14.) If f(x) = , then f’(x) = 1 cos x
(A) cos x (B) sin x (C) –sin x (D) –cos x (E) 2 cos x (HINT: Since sin 2 x cos 2 x 1, then sin 2 x 1 cos 2 x which can be factored.)
15.)* At the point of intersection of f(x) = cos x and g(x) = 1 - x 2 , the tangent lines are:
(A) the same line (B) parallel lines (C) perpendicular lines (D) intersecting but not perpendicular lines (E) none of the above (HINT: Use graph-isect to find x value of intersection point).
2 3 16.)* The tangent line to the graph of y = sin x at ( , ) crosses y = sin x at the 3 2 point where x = _____.
(A) -.781 (B) 4.712 (C) 5.388 (D) 5.760 (E) 6.283 (HINT: Find tangent line equation and graph with y = sin x…use isect.)
17.) What is the 50th derivative of cos x? (A) – cos x (B) cos x (C) sin x (D) –sin x (E) 0
18.) Find the x-coordinates of all points, -2 < x < 2, where the line y = x + b is tangent to the graph of f(x) = x + b(sin x). (HINT: The slope of y = x + b must match derivative of y = x + b sin x for it to be tangent line.) (This question is from the 1996 exam: FR #4b.)
Assignment 8: The Chain Rule
In 1-4, find y’ using the Chain Rule. 1.) y = cos (x 2 ) 2.) y = sin 2 x 3.) y = 6 tan 3 x 4.) y = cos (x 3 - 2x)
In 5 and 6, find y’ by “double Chain-Ruling”. 5.) y = sin 3 (x 2 ) 6.) y = 3(sin(x 2 ) + tan(x 3 ))5
In 7-10, find y’ by using the Chain Rule in conjunction with the Product/Quotient Rules. 7.) y = sin 2 x cos(x 2 ) 8.) y = ( x x)(cot(3x)) 3cos(2x) tan( x) 9.) y = 10.) y = 2sin(3x) 3sin 2 x
dy 11.) Find if y = f(b) and b = g(x). dx
12.) Use the data from the table to evaluate the derivative of each function at the given x value. x f(x) g(x) f’(x) g’(x) 1 2 8 2 -3 3 3 3 -4 2 5
a.) 2f(x) at x = 2 b.) f(x) + g(x) at x = 3 f (x) c.) f(x) g(x) at x = 3 d.) at x = 2 g(x) e.) f(g(x)) at x = 2 f.) f (x) at x = 2 1 g.) at x = 3 h.) f 2 (x) g 2 (x) at x = 2 g 2 (x)
tan 2 (x h) tan 2 x 13.) Evaluate the limit: lim h 0 h
14.) Find the equation of each horizontal tangent line of y = sin x on [0, 2 ] and state the x value where each occurs.
Assignment 8B: Using the Chain Rule 1.) A normal line to the graph of a function f at the point (x, f(x)) is defined to be the line perpendicular to the tangent line at that point. The equation of the normal line to the curve y = 3 x 2 1 at the point where x = 3 is:
(A) y + 12x = 38 (B) y – 4x = 10 (C) y + 2x = 4 (D) y + 2x = 8 (E) y – 2x = -4
2.) The derivative of 4x3 2x6 is:
(A) 72x 8 (B) 124x 17 (C) 30x(4x) 2 (2x) 5 (D) 72x(4x) 2 (2x) 5 (E) 144(4x) 2 (2x) 5
3.) If f(x) = (x-1) 2 cos x, then f’(0) =
(A) –2 (B) –1 (C) 0 (D) 1 (E) 2
4.) If f(x) = 4sin x 2 , then f’(0) = 2 (A) –2 (B) 0 (C) 2 (D) (E) 1 2
5.) What is the 20th derivative of y = sin (2x)?
(A) - 220 sin(2x) (B) 220 sin(2x) (C) - 219 cos(2x) (D) 220 cos(2x) (E) 221 cos(2x)
6.) The equation of the tangent line to the curve x 2 + y 2 = 169 at the point (5, -12) is
(A) 5y – 12x = -120 (B) 5x – 12y = 119 (C) 5x – 12y = 169 (D) 12x + 5y = 0 (E) 12x + 5y = 169 1 2 2 (HINT: Let y = - 169 x or y = - (169 – x ) 2 )
7.) Let f(x) be a continuous and differentiable function. The table below gives the values of f(x) and 1 f’(x), the derivative of f(x), at several values. If g(x) = , what is the value of g’(2)? f (x) X 1 2 3 4 F(x) -3 -8 -9 0 F’(x) -5 -4 3 16
1 1 1 (A) - (B) 0 (C) (D) (E) 16 8 16 64 (HINT: Use the chain rule to find the derivative of g (x) = f (x)1 . x 8.) Let f be the function given by f(x) = . (1989 exam: FR #4). x 2 4 a. Find the domain of f. (HINT: What value is not acceptable for x?) b. Write an equation for each vertical asymptote to the graph of f. c. Find f ’(x).
9.) Let f(x) = 1 sin x . (1987 exam: FR #2.) a. Find the domain of f. b. Find f ‘(x). c. What is the domain of f ‘(x)? d. Write an equation for the line tangent to the graph of fat x = 0.
Assignment 9: Implicit Differentiation
dy In 1-4, find using implicit differentiation. dx
1.) y2 + x2 = 1
2.) sin y + xy = x
3.) y3 – 2xy = cos (xy)
4.) y2 + 6xy – 6x = 3y
d 2 y 5.) Find if y2 + xy = x + y dx2
6.) For x2 – xy + y2 = 7: a. Find the equation of the tangent line at (-1, 2). b. Find the equation of the normal line at (-1, 2).
7.) If 2x3 – 3y2 = 16, find the equation of the curve’s only vertical tangent line.
8.) If x2 + 4xy = -75 – y2, find the equation of the curve’s horizontal tangent lines and state the x values at which each one occurs.
9.) Find the indicated higher-order derivative: a. y’’ if y = sin2 x b. y’’’ if y = cos (x4) c. y(20) if y = sin (3x) d. y’’ if y = tan x Assignment 9b: Implicit Differentiation
dy 1.) If x + y = xy, then is dx 1 y 1 1 y 2 xy (A) (B) (C) (D) x + y – 1 (E) x 1 x 1 x 1 y
d 2 y 2.) If x 2 y + yx 2 = 6, then at the point (1, 3) is dx 2 (A) –18 (B) –6 (C) 6 (D) 12 (E) 18
dy 3.) If sin (xy) = x 2 , then = dx sec(xy) 2xsec(xy) (A) 2x sec(xy) (B) (C) 2x sec (xy) – y (D) x 2 y 2xsec(xy) y (E) x
dy 4.) If y 2 -2xy = 21, then at the point (2, -3) is dx 6 3 2 3 3 (A) - (B) - (C) - (D) (E) 5 5 5 8 5
5.) Consider the curve given by xy2 – x3y = 6. (This question is from the 2000 exam: FR #5.)
dy 3x 2 y y 2 a. Show that . dx 2xy x 3
b. Find all points on the curve whose x-coordinate is 1, and write an equation for the tangent line at each of these points. (HINT: Let x = 1 and solve the original equation for y. You should get two solution which you can use to find the slope of each tangent line.)
c. Find the x-coordinate of each point on the curve where the tangent line is vertical. (HINT: What is the slope of a vertical tangent line?)
6.)* Consider the curve defined by –8x2 = 5xy + y3 = -149. (This question is from the 1995 exam: FR #3.) dy a. Find . dx b. Write an equation for the line tangent to the curve at the point (4, -1).
c. There is a number k so that the point (4.2, k) is on the curve. Using the tangent line found in part b, approximate the value of k.
d. Write an equation that can be solved to find the actual value of k so that the point (4.2, k) is on the cure.
e. Solve the equation in found in part d for the value of k.
7.) Consider the curve defined by x2 + xy + y2 = 27. (This question is from the 1994 exam: #3.)
a. Write an expression for the slope of the curve at any point (x, y).
b. Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel.
c. Find the points on the curve where the lines tangent to the curve are vertical.
d. 8) Consider the curve defined by the equation: y + cos y = x + 1. (This question was on the 1992 exam: FR# 4.)
dy e. a) Find in terms of y. dx
e. b) Write an equation for each vertical tangent to the curve. d 2 y f. C) Find in terms of y. dx 2
Assignment 10: Inverse Derivatives
In 1-3, given f(x), find the derivative of f –1(x) at the indicated point. The indicated point is on the graph of the inverse.
1.) f (x) = x2 + 3x + 2 at (2, 0) 1 2.) f (x) = sin x at , 2 3 x 1 3.) f (x) = cos2 at , 4 2
4.) If f (x) = g-1 (x) and f (2) = 3 and f’ (2) = 5, find g’ (3).
5.) If m (t) and Q (t) are inverses and if Q (7) = 3 and Q’ (3) = 10 and Q’ (7) = 15, find m’ (3). 6) 90, 40
7) 154, 24
8) 185, 14