Find the derivative of the function. F(t) = e t sin 2t
> In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential function P(t) = (1436.53) . (1.01395)t where t = 0 corresponds to the year 1900 and Pstd is measured in millions. According to this model, what was the rate of increase
> The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function C(t) = 0.0225te-0.0467t where t
> Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. Ка) f(x) — х" (b) f(x) = 1/x
> The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on suc
> If the equation of motion of a particle is given by s = A cos (ωt + δ), the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0?
> The displacement of a particle on a vibrating string is given by the equation s(t) = 10 + 1/4 sin(10πt) where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds.
> Let f (x) = cx + ln (cos x). For what value of c is f ‘( π/4) = 6?
> Find the 1000th derivative of f (x)=xe-x.
> For what values of r does the function y=erx satisfy the differential equation y’’ - 4y’ + y=0?
> Show that the function y=e2x (A cos 3x + B sin 3x) satisfies the differential equation y’’ - 4y’ + 13y=0.
> If F(x) = f (x f (x f (x))), where f (1) = 2, f (2) = 3, f ‘(1) = 4, f ‘(2) = 5, and f ‘(3) = 6, find f’(1).
> If F(x) = f (3 f(4 f (x))), where f (0) = 0 and f ‘(0) = 2, find F’(0).
> If g is a twice differentiable function and f (x) = xg(x2), find f ‘’ in terms of g, g’, and g’’.
> Let r(x) = f (g(h(x))) , where h(1) = 2, g(2) = 3, h’(1) = 4, g’(2) = 5, and f ‘(3) = 6. Find r’(1).
> 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. 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. f(t)= t sin πt
> 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
