Find the derivative of the function. f(t)= t sin πt
> Let g(x) = ecx + f (x) and h(x) = ekx f (x), where f (0) = 3, f ‘(0) = 5, and f ‘’(0) = 22. (a) Find g ‘(0) and g ‘‘(0) in terms of c. (b) In terms of k, find an equation of the tangent line to the graph of h at the point where x − 0.
> Find equations of the tangent lines to the curve y= ln x/x at the points (1, 0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines.
> Suppose f is differentiable on R. Let F(x) − f (ex) and G(x) = ef (x). Find expressions for (a) F ‘(x) and (b) G ‘(x).
> Suppose f is differentiable on R and α is a real number. Let F(x) = f (ex) and G(x) = [f(x)]α . Find expressions for (a) F’ (x) and (b) G’ (x).
> Differentiate. f (x) − x2 sin x
> If g(x) = f(x) , where the graph of f is shown, evaluate g’ (3). |f/ -1 1
> If g is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x g(x) (b) y = x / g(x) (c) y = g(x) / x
> If f is the function whose graph is shown, let h(x) =f(f(x)) and g(x) = f(x2). Use the graph of f to estimate the value of each derivative. (a) h’ (2) (b) g’ (2) y= f(x) 1 - 이 1
> If f and g are the functions whose graphs are shown, let u(x) = f (g(x), v(x) = g(f(x), and w(x) = g(g(x). Find each derivative, if it exists. If it does not exist, explain why. (a) u’(1) (b) v’(1) (c) wâ€&#
> Let f and g be the functions in Exercise 63. (a) If F(x) = f (f(x)), find F’(2). (b) If G(x) = g(g(x), find G’(3). Data from Exercise 63: A table of values for f, g, f ‘, and g ‘ is
> A table of values for f, g, f ‘, and g ‘ is given. (a) If h(x) = f (g(x)), find h’(1). (b) If H(x) = g(f(x)), find H’(1). f(x) g(x) f'(x) g'(x) 1 4 1 8 5 3 7 2 7 679 3. 2.
> If F(x) = f (g(x)), where f (-2) = 8, f ‘ (-2) = 4, f ‘ (5) = 3, g(5) = -2, and g’(5) = 6, find F’ (5).
> If f (x) = sin x + ln x, find f 9sxd. Check that your answer is reasonable by comparing the graphs of f and f ‘.
> Find all points on the graph of the function f (x) = 2 sin x + sin2x at which the tangent line is horizontal.
> The function f (x) = sin (x + sin 2x), 0 ≤ x ≤ π, arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a calculator to make a rough sketch of the graph of f ‘. (b) Calculate f ‘(x) and use this expression, with
> 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). F G 1
> (a) The curve y = |x |/ 2 – x 2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (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 = 2 / (1 + e-x) at the point (0, 1). (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 = sin (sin x) , (π , 0)
> Find an equation of the tangent line to the curve at the given point. y = x2 ln x
> Find an equation of the tangent line to the curve at the given point. y = 2x, (0 , 1)
> Find y’ and y’’. y = cos (sin θ)
> If f and t are the functions whose graphs are shown, let u(x) = f (x)g(x) and v(x) = f (x)/g(x). (a) Find u’(1). (b) Find v’(5). f 1 1
> Find the derivative of the function. y = [x+(x + sin2 x)3]4
> Find the derivative of the function. g(x) = (2rarx + n)p
> Find an equation of the tangent line to the curve at the given point. y = ln (x2 – 3x + 1)
> Find the derivative of the function. f(1) = sin²(e*n*r)
> Find the derivative of the function. y = e sin 2x + sin (e2x)
> Find the derivative of the function. f(t) = tan (sec(cos t))
> Find the derivative of the function. y = cot2 (sin θ)
> If f (2) = 10 and f ‘(x)= x2 f (x) for all x, find f ‘’(2).
> Find the derivative of the function. y = x2 e-1/x
> Differentiate each trigonometric identity to obtain a new (or familiar) identity. (a) tan x = sin x / cos x (b) sec x = 1 / cos x (c) sin x + cos x = 1 + cot x / csc x
> If f (x) = cos(ln x2), find f ‘(1).
> Find the derivative of the function. G(x) = 4 C/x
> Find the derivative of the function. F(t) = e t sin 2t
> Find the derivative of the function. J (θ) = tan2 (nθ)
> If g(x) = x f (x), where f (3) = 4 and f ‘(3) = -2, find an equation of the tangent line to the graph of g at the point where x = 3.
> If f (x) = ln(x + ln x), find f ‘(1).
> Find the derivative of the function. f(t) = 2 t 3
> Find the derivative of the function. Y = etan θ
> Find the derivative of the function. Y = (x + 1/x)5
> Find the derivative of the function. F(t) = (3t – 1)4 (2t +1)-3
> Find the derivative of the function. h(t) = (t + 1)2/3 (2t2 – 1)3
> Find the derivative of the function. g(x) = (x2 + 1)3 (x2 + 2)6
> Find the derivative of the function. f(x) = (2x - 3)4 (x2 + x + 1)5
> If h(2) = 4 and h’(2) = -3, find d h(x) dx I-2
> Differentiate f and find the domain of f. f(x) = ln ln ln x
> Find the derivative of the function. f(t) = eat sin bt
> Find the derivative of the function. Y = x2 e-3x
> Find the derivative of the function. g (θ) = cos2 (θ)
> Find the derivative of the function. f (θ) = cos (θ2)
> Suppose that f(4) = 2, g(4) = 5, f ‘ (4) = 6 and g’ (4) = -3. Find h ‘ (4). (a) h(x) = 3f (x) + 8g(x) (b) h(x) = f(x) g(x) (c) h(x) = f(x) / g(x) (d) h(x) = g(x) / f(x) + g(x)
> Find the derivative of the function. F(x) = (1 + x + x2)99
> Find the derivative of the function. F(x) = (5x6 + 2x3)4
> If f (x) = ex g(x), where g(o) = 2 and g’(o) = 5, find f ‘(0).
> Differentiate f and find the domain of f. f(x) = ln (x2 – 2x)
> Differentiate. y = 2 sec x - csc x
> Differentiate. f (x) = ex cos x
> If f is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x2 f (x) (b) y = f (x) / x2 (c) y = x2 / f (x) (d) y = 1 + x f (x) / √x
> Differentiate. f (x) = x cos x + 2 tan x
> Find the 50th derivative of y=cos 2x.
> Differentiate the function. H(u) = (3u - 1)(u + 2)
> Differentiate. f (x) = (3x2 - 5x)ex
> Suppose that f (5) = 1, f ‘(5) = 6, g(5) = -3, and g’ (5) = 2. Find the following values. (a) (fg) ‘ (5) (b) (f/g) ‘ (5) (c) (g/f) ‘ (5)
> 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. If f is differentiable and f (-1) = f (1), then there is a number c such that |c | < 1 and f ‘(c) =
> 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. If f is continuous on (a, b), then f attains an absolute maximum value f (c) and an absolute minimu
> 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. If f has an absolute minimum value at c, then f ‘(c) = 0.
> 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. If f ‘(c) = 0, then f has a local maximum or minimum at c.
> 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. If f ‘(x) exists and is nonzero for all x, then f (1) ≠ f (0).
> 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. The most general antiderivative of f sxd − x22 is F(x) = -1/x + C
> 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. If f is periodic, then f ’ is periodic.
> 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. If f is even, then f ’ is even.
> 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. If f is increasing and f (x) > 0 on I, then g(x) = 1/f (x) is decreasing on 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. If f and t are positive increasing functions on an interval I, then f g is increasing on 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. If f and t are increasing on an interval I, then f g is increasing on 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. If f and t are increasing on an interval I, then f - g is increasing on 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. If f and t are increasing on an interval I, then f + g is increasing on I.
> Find y’ and y’’. y = ln (1 + ln x)
> 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. There exists a function f such that f (x) < 0, f ‘(x) < 0, and f ’’(x) > 0 for all x.
> 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. There exists a function f such that f (x) > 0, f ‘(x) < 0, and f ’’ (x) > 0 for all x.
> 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. There exists a function f such that f (1) = -2, f (3) = 0, and f ‘(x) > 1 for all x.
> 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. If f ‘(x) = g’(x) for 0 < x < 1, then f (x) = g(x) for 0 < x < 1.
> 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. If f ‘’(2) = 0, then (2, f (2)) is an inflection point of the curve y = f (x).
> 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. If f ‘(x) < 0 for 1 < x < 6, then f is decreasing on (1, 6).
> In this project we investigate the most economical shape for a can. We first interpret this to mean that the volume V of a cylindrical can is given and we need to find the height h and radius r that minimize the cost of the metal to make the can (see the
> Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain t
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = x(12x + 8)
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 6x5 - 8x4 - 9x2
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 2x3 – 2/3x2 + 5x
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = x2 - 3x + 2
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 4x + 7
> A high-speed bullet train accelerates and decelerates at the rate of 4 ft/s2. Its maximum cruising speed is 90 mi/h. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at th
> A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a(t) = 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (do
> A car is traveling at 100 km/h when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?
> A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?
> What constant acceleration is required to increase the speed of a car from 30 mi/h to 50 mi/h in 5 seconds?
> A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant deceleration of 22 ft/s2. What is the distance traveled before the car comes to a stop?
> Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m/s and its downward acceleration is If the raindrop is initially 500 m above the g
> A company estimates that the marginal cost (in dollars per item) of producing x items is 1.92 - 0.002x. If the cost of producing one item is $562, find the cost of producing 100 items.
