(a). If F (x) = f (x) g (x), where f and g have derivatives of all orders, show that F" = f"g +2f'g' + fg". (b). Find similar formulas for f"' and F(4). (c). Guess a formula for F(R).
> Find the derivative of the function. y = 3 cot (nθ)
> Find the derivative of the function. h (t) = t3 - 3t
> Find the derivative of the function. y = a3 + cos3x
> Find the derivative of the function. y = cos (a3 + x3)
> Find the derivative of the function. f (t) = 3√1 + tan t
> Find the derivative of the function. f (z) = 1/z2 + 1
> Find equations of the tangent line and normal line to the given curve at the specified point. y = 2xex, (0, 0)
> Find the derivative of the function. f (x) = (1 + x4)2/3
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx.
> Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = sin x
> The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function M' (t). During which years was the derivative negative?
> Differentiate the function. B (x) = cy-6
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. if limx→ 6 [f (x) g (x)] exists, then the limit must be f (6) g (6). 7. I
> Determine whether the statement is true or false. If it is true, explain why If it is false, explain why or give an example that disproves the statement. 1. If f and g are differentiable, then 2. If f and g are differentiable, then 3. If f and g are
> Differentiate. y = x/2 – tan x
> Differentiate the function. h (x) = (x – 2) (2x + 3)
> Use the given graph to estimate the value of each derivative. Then sketch the graph of f'. (a). f'(-3) (b). f'(-2) (c). f'(-1) (d). f'(0) (e). f'(1) (f). f'(2) (g). f'(3)
> If R denotes the reaction of the body to some stimulus of strength x, the sensitivity is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts
> On what interval is the function f (x) = 5x - ex increasing?
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = x + 1/x
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = 3x15 - 5x3 + 3
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = ex - 5x
> Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = x - √x (1, 0)
> Differentiate. f (t) = cot t/et
> Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = 3x2 - x3 (1, 2)
> Find equations of the tangent line and normal line to the curve at the given point. y = x4 + 2e (0, 2)
> Find an equation of the tangent line to the curve at the given point. y = x4 + 2x2 – x, (1, 2)
> Find an equation of the tangent line to the curve at the given point. y = 4√x, (1, 1)
> For what values of does the graph of have a horizontal tangent? f (x) = x + 2 sin x
> Differentiate the function. y = ex+1 + 1
> Differentiate the function. z = A/y10 + Bey
> Differentiate the function. v = (√x + 1/√x)2
> Evaluate limx→1 x1000 – 1/x - 1
> Differentiate the function. u = 5√t + 4√t5
> Differentiate. y = c cos t+ t2 sin t
> Differentiate the function. y = aev + b/v + c/v2
> Find the value of such that the line y = 3/2 + 6x is tangent to the curve y = c√x.
> Differentiate the function. y = 4π2
> Differentiate the function. g (u) = √2 u + √3 u
> (a). The curve with equation y2 = x3 – 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, -2). (b). At what points does this curve have horizontal tangents? (c). Illustrate parts (a) and (b) by grap
> The graph indicates how Australia’s population is aging by showing the past and projected percentage of the population aged 65 and over. Use a linear approximation to predict the percentage of the population that will be 65 and over in
> Differentiate the function. y = x2 + 4x +3/√x
> Use the result of Exercise 63(c) to find an antiderivative of each function. (a). f (x) = √x (b). f (x) = ex + 8x3 Exercise 63(c): (c). Find an antiderivative for f (x) xn, where n ≠ -1. Check by differentiation.
> Differentiate the function. f (x) = x2 -3x + 1/x2
> The equation y" + y' -2y = x2 is called a differential equation because it involves an unknown function and its derivatives y' and y". Find constants A, B, and C such that the function y = Ax2 + BX + C satisfies this equation. (Differential equations wil
> Find a second-degree polynomial P such that P (2) = 5, P'(2) = 3, and P"(2) = 2.
> (a). If g is differentiable, the Reciprocal Rule says that Use the Quotient Rule to prove the Reciprocal Rule. (b). Use the Reciprocal Rule to differentiate the function in Exercise 16. Exercise 16: Differentiate y = 1/s + kes (c). Use the Reciproc
> Differentiate. g (θ) = eθ (tan θ – θ)
> Use the definition of a derivative to show that if f (x) = 1/x, then f'(x) = -1/x2. (This proves the Power Rule for the case n = -1.)
> (a). Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh + fg'h + fgh'. (b). Taking f = g = h in part (a), show that d/ dx [f (x)]3 = 3[f (x)]2 f'(x) (c). Use part (b) to differentiate y = e3x.
> The cost, in dollars, of producing yards of a certain fabric is C (x) = 1200 + 12x – 0.1x2 + 0.0005x3 (a). Find the marginal cost function. (b). Find C"(200) and explain its meaning. What does it predict? (c). Compare C"(200) with the cost of manufacturi
> Use the method of Exercise 55 to compute Q'(0), where Q (x) = 1 + x + x2 + xex/ 1 – x + x2 - xex Exercise 55: Find R'(0), R(x) = x + 3x3 + 5x5/1 + 3x3 + 6x6 + 9x9 where Hint: Instead of finding R’(x) first, let f (x) be the numerator and g (x) the deno
> At what point on the curve y = 1 + 2ex – 3x is the tangent line parallel to the line 3x – y = 5? Illustrate by graphing the curve and both lines.
> How many tangent lines to the curve y = x/ (x + 1) pass through the point (1, 2)? At which points do these tangent lines touch the curve?
> On what interval is the function f (x) = x2ex concave downward?
> On what interval is the function f (x) = x3ex increasing?
> (a). If g (x) = x2/3, show that g'(0) does not exist. (b). If a ≠ 0, find g'(a). (c). Show that y = x2/3 has a vertical tangent line at (0, 0). (d). Illustrate part (c) by graphing y = x2/3.
> Differentiate. y = sec θ tan θ
> Find the points on the curve y = 2x3 + 3x2 – 12x + 1 where the tangent is horizontal.
> Differentiate each trigonometric identity to obtain a new (or familiar) identity.
> Let P (x) = F (x) G (x) and Q (x) = F (x)/ G (x), where F and G are the functions whose graphs are shown. (a). Find p'(2). (b). Find Q'(7).
> The frequency of vibrations of a vibrating violin string is given by where L is the length of the string, T is its tension, and is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA, 2002).] (a). Find the r
> Use Formula 2 and trigonometric identities to evaluate the limit. limθ→0 sin θ/θ + tan θ
> Use Formula 2 and trigonometric identities to evaluate the limit. limx→0 sin 3x sin 5x/ x2
> Use Formula 2 and trigonometric identities to evaluate the limit. limt→0 tan 6t/sin 2t
> (a). Use the substitution θ = 5x to evaluate limx→0 sin 5x/x (b). Use part (a) and the definition of a derivative to find d/dx (sin 5x)
> Suppose that f (5) = 1, f'(5) = 6, g (5) = -3, and g'(5) = 2. Find the following values.
> Find the given derivative by finding the first few derivatives and observing the pattern that occurs.
> Differentiate. f (x) = √x sin x
> Find the given derivative by finding the first few derivatives and observing the pattern that occurs.
> (a). If f (x) = (x2 – 1) ex, find f'(x) and f"(x). (b). Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f', and f".
> A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x cha
> The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of th
> (a). If f (x) = ex/ (2x2 + x + 1), find f'(x). (b). Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
> A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x (t) = 8 sin t, where is in seconds and in centimeters. (a). Find the velocity and acceleration at time t. (b). Find the position, velocity,
> (a). The curve y = x/ (1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3,0.3). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Let f (x) = x – 2 sin x, 0 < x < 2π. On what interval is f increasing?
> For what values of does the graph of have a horizontal tangent? f (x) = ex cos x
> (a). Use the definition of derivative to calculate f'. (b). Check to see that your answer is reasonable by comparing the graphs of f and f'. f (x) = x4 + 2x
> Suppose f (π/3) = 4 and f'(π/3) = -2, and let g (x) = f (x) sin x and h (x) = (cos x)/f (x). Find (a). g'(π/3) (b). h'(π/3)
> Differentiate. f (x) = sinx + 1/2 cot x
> (a). Use the Quotient Rule to differentiate the function f (x) = tan x- 1/sec x (b). Simplify the expression for f (x) by writing it in terms of sin x and cos x, and then find f'(x). (c). Show that your answers to parts (a) and (b) are equivalent.
> If f (t) = csc t, find f"(π/6).
> Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it.
> (a). If f (x) = ex cos x, find f'(x) and f"(x). (b). Check to see that your answers to part (a) are reasonable by graphing f, f', and f".
> (a). If f (x) = sec x - x, find f'(x). (b). Check to see that your answer to part (a) is reasonable by graphing both f and f' for |x| < π/2.
> (a). Find an equation of the tangent line to the curve y = 3x + 6 cos x at the point (π/3, π + 3). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> (a). Find an equation of the tangent line to the curve y = 2x sin x at the point (π/2, π). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Find an equation of the tangent line to the curve at the given point. y = 1/sin x + cos x, (0, 1)
> Find an equation of the tangent line to the curve at the given point. y = x + c0s x, (0, 1)
> Find an equation of the tangent line to the curve at the given point. y = ex cox, (0, 1)
> Differentiate. y = 2csc x + 5 cos x
> Find an equation of the tangent line to the curve at the given point. y = sec x, (π/3, 2)
> Prove, using the definition of derivative, that if f (x) = csc x, then f'(x) = -sin x.
> The table gives the population of the world in the 20th century. (a). Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b). Use a graphing calculator or computer to find a cubic function (a third-d
> Prove that d/dx (cot x) = -csc2x.
> Prove that d/dx (sec x) = sec x tan x.
> Prove that d/dx (csc x) = -csc x cot x.
> Differentiate. y = x2 sin x tan x
> Differentiate. f (x) = xex csc x
