Differentiate the function. f (x) = x3 – 4x + 6
> Sketch the graph of a function that satisfies all of the given conditions.
> Differentiate. f (t) = 2t/ 4 + t2
> Differentiate. g (x) = 3x – 1/2x + 1
> Differentiate. y = ex/1 + x
> Differentiate. y = ex/x2
> If f is a differentiable function, find an expression for the derivative of each of the following functions.
> If is a differentiable function, find an expression for the derivative of each of the following functions.
> Use the definition of a derivative to find f'(x) and f''(x). Then graph f, f', and f'' on a common screen and check to see if your answers are reasonable. f (x) = 3x2 + 2x + 1
> If h (2) = 4 and h'(2) = -3, find d/dx (h (x)/x) |x-2
> If f (x) = exg, where g (0) = 2 and g'(0) = 5, find f'(0).
> Suppose that f (2) = -3, g (2) = 4, f'(2) = -2, and. g'(2) Find h'(2).
> In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number of bacteria after hours and use it to estimate the ra
> If g (x) = x/ex, find g(n)(x).
> Differentiate. g (x) = √x ex
> If f (x) = x2/ (1 + x), find f"(1).
> The graph of f is given. State, with reasons, the numbers at which f is not differentiable.
> (a). If f (x) = (x3 – x) ex, find f'(x). (b). Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
> (a). The curve y = 1/ (1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point (-1, 1/2). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Find equations of the tangent line and normal line to the given curve at the specified point. y = √x/ x + 1, (4, 0.4)
> Find an equation of the tangent line to the given curve at the specified point. y = ex/x, (1, e)
> Differentiate. f (x) = (x3 + 2x) ex
> Find an equation of the tangent line to the given curve at the specified point. y = 2x/x + 1, (1, 1)
> Let f (x) = (x – 1)2 g (x) = e-2x and h (x) = 1 + ln (1 – 2x) (a). Find the linearizations of f, g, and h at a = 0. What do you notice? How do you explain what happened? (b). Graph f, g, and h and their linear approximations. For which function is the li
> Find f"(x) and f"(x). f (x) = x/x2 - 1
> Find f"(x) and f"(x). f (x) = x2 / 1 + 2x
> Find f"(x) and f"(x). f (x) = x5/2 ex
> Find f"(x) and f"(x). f (x) = x4 ex
> Differentiate. f (x) = ax + b/cx + d
> Differentiate. f (x) = x/x + c/x
> Differentiate. f (x) = 1 – xex/x + ex
> Differentiate. f (x) = A/B + Cex
> Differentiate. g (t) = t - √t/t1/3
> Find the derivative of the function in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?
> Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV = C. (a). Find the rate of change of volume with respect to pressure. (b). A sample of gas is in a conta
> Differentiate. f (t) = 2t/2 + √t
> Differentiate. z = w3/2 (w + cew)
> Differentiate. y = v3 – 2v/v/v
> Differentiate. y = 1/s + kex
> Differentiate. y = (r2 - 2r) er
> Differentiate. y = t/ (t – 1)2
> Differentiate. y = t2 + 2/t4 – 3t2 + 1
> Differentiate. y = x + 1/ x3 + x -2
> Differentiate. y = x3/1 – x2
> Differentiate. R (t) = (t + et) (3 - √t)
> Newton’s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is F = GmM/r2 where G is the gravitational constant and r is the distance between the bodies. (a). Find dF/dr and explain its meaning. What
> A particle moves according to a law of motion s = f (t), t > 0, where is measured in seconds and in feet. (a). Find the velocity at time t. (b). What is the velocity after 3 s? (c). When is the particle at rest? (d). When is the particle moving in the po
> Find the derivative of f (x) = (1 + 2x2) (x – x2) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?
> Differentiate the function. f (t) = 1/4 (t4 + 8)
> Differentiate the function. f (t) = 1/2t6 – 3t4 + t
> Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. (a). f (x) = xn (b). f (x) = 1/x
> Differentiate the function. F (x) = 3/4x8
> Differentiate the function. F (x) = (1/2x)5
> Differentiate the function. y = √x (x - 1)
> Differentiate the function. y = 3ex + 4/3√x
> Find an equation of the normal line to the parabola y = x2 – 5x + 4 that is parallel to the line x – 3y = 5.
> The quantity of charge in Q coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Q (t) = t3 – 2t2 + 6t + 2. Find the current when (a) t = 0.5s and (b) t = 1s. [See Example 3. The unit of current is an ampe
> Differentiate the function. h (t) = 4√t – 4et
> Differentiate the function. g (t) = 2t-3/4
> Show that the function f (x) = |x – 6| is not differentiable at 6. Find a formula for f' and sketch its graph.
> Differentiate the function. f (t) = 2 – 2/3t
> On what interval is the function f (x) = x3 – 4x2 + 5x concave upward?
> If f (x) = 2x2 – x2, find f'(x), f'''(x), f'''(x), and f4(x). Graph f, f', f'', and f" on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?
> The equation of motion of a particle is s = t4 – 2t3 + t2 - t, where s is in meters and t is in seconds. (a). Find the velocity and acceleration as functions of t. (b). Find the acceleration after 1 s. (c). Graph the position, velocity, and acceleration
> Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f', and f". f (x) = ex - x3
> Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f', and f". f (x) = 2x - 5x3/4
> Find the first and second derivatives of the function. G (r) = √r + 3√r
> Use a linear approximation (or differentials) to estimate the given number. 1/1002
> Find the first and second derivatives of the function. f (x) = 10x10 + 5x5 - x
> (a). Use a graphing calculator or computer to graph the function g (x) = ex – 3x2 in the viewing rectangle [-1, 4] by [-8, 8]. (b). Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of g'. (See Example 1 in Sectio
> Differentiate the function. f (x) = √30
> (a). How is the number e defined? (b). Use a calculator to estimate the values of the limits correct to two decimal places. What can you conclude about the value of e?
> The graph of f is given. State, with reasons, the numbers at which f is not differentiable.
> The graph of f is given. State, with reasons, the numbers at which f is not differentiable.
> The graph of f is given. State, with reasons, the numbers at which f is not differentiable.
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = 3x5 - 20x3 + 50x
> The unemployment rate U (t) varies with time. The table (from the Bureau of Labor Statistics) gives the percentage of unemployed in the US labor force from 1998 to 2007. (a). What is the meaning of U' (t)? What are its units? (b). Construct a table of
> (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 (t) = t2 - √t
> (a). Sketch, by hand, the graph of the function f (x) = ex, paying particular attention to how the graph crosses the y-axis. What fact allows you to do this? (b). What types of functions are f (x) = ex and g (x) = xe? Compare the differentiation formulas
> (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) = x + 1/x
> Differentiate the function. f (x) = 186.5
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = x4
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = x +√ x
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = x2 - 2x3
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = 1.5x2 - x + 3.7
> Use a linear approximation (or differentials) to estimate the given number. e-0.015
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (t) = 5t - 9t2
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f (x) = mx + b
> Use the given graph to estimate the value of each derivative. Then sketch the graph of f'. (a). f'(0) (b). f'(1) (c). f'(2) (d). (3) (e). f'(4) (f). f'(5) (g). f'(6) (h). f'(7)
> Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
> Let f (x) = x3. (a). Estimate the values of f'(0), f'(1/2), f'(1), f'(2), and f'(3) by using a graphing device to zoom in on the graph of f. (b). Use symmetry to deduce the values of f'(-1/2), f'(-1), f'(-2), and f'(-3). (c). Use the values from parts (a
> Let f (x) = x2. (a). Estimate the values of f'(0), f'(1/2), f'(2), and by using a graphing device to zoom in on the graph of f. (b). Use symmetry to deduce the values of f'(-1/2), f'(-1), and f'(-2). (c). Use the results from parts (a) and (b) to guess a
> 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) = ln x
> 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) = ex
> If y = f (u) and u = g (x), where f and g are twice differentiable functions, show that
> (a). Write |x| = √x2 and use the Chain Rule to show that d/dx |x| = x/|x| (b). If f (x) = |sin x|, find f'(x) and sketch the graphs of f and f'. Where is f not differentiable? (c). If g (x) = sin |x|, find g'(x) and sketch the graphs of g and g'. Where i
> The graph of the derivative f' of a continuous function f is shown. (a). On what intervals f is increasing? Decreasing? (b). At what values of x does f have a local maximum? Local minimum? (c). On what intervals is f concave upward? Concave downward? (
