A) If x, y > 0, then ln x - ln y = ln x ¯¯¯¯ ln y b) If f'(a) = 0 and f"(a) = 0, then the function f does not have an extreme point at x = a. c) For every real number x, we have ln(e^x²-¹) = (x - 1)(x + 1) (e
Find and simplify derivatives of the functions: a) y = cos x ¯¯¯¯¯¯¯ x^2+e^2x b) y = ln((x^2 + 1)(cos x)) (^ means exponent and e^2x, 2x is the exponent)
Use chain rule to verify that every function of the form y = a sin (5t) + b cos (5t) is a solution to the differential equation d^2y/dt^2 = -25y. Then use this fact to find the solution which also satisfies the initial conditions: y(0) = 3 and y'(0) = 0 (^ means exponent and d^2y is over dt^2)
Suppose f(x) = ln(2 + cos x) on the interval (0, 2pi). a) Calculate f' (x) and f" (x). b) Find the interval(s) on which the function f is increasing. c) Find all extreme values of f and the values of x at which they occur. d) Find the interval(s) on which the function f is concave up?
Please explain the steps and solution, thanks: The equation 4x^2y - 3y = x^3 implicitly defines y as a function of x. a) use implicit differentiation to find dy/dx. b) write y as an explicit function of x and compute dy/dx directly.
Use L'Hopital's rule to evaluate the limit lim e^x - x - 1 x→0 ¯¯¯¯¯¯¯¯¯ x^2
Find and simplify the derivatives of the functions: a) k(x) = sin^7(2x) b) g(x) = xe^2x (^ means exponent and ^2x is the total exponent of xe)
Calculate derivatives of the function f (x) = 2^x + x^2 + pi^2 + ln2
Please explain the steps and solution: Calculate derivatives of the function h (x) = ln (x^3) - ln (^2) (^ means exponent)
Calculate the derivatives of the function: g (x) = x^2 - 5x/ √x (x^2 - 5x is over √x as in a fraction and ^ means exponent)
Calculate the derivatives of the functions: a) k (x) = 3^4√x^5 - 5/^3√x^2 (this means it is a fraction: 5 is over ^3√x^2)
Please explain the steps and solution, thank you: Find two functions f and g such that the derivatives f' (x) = g' (x) = 3x^2 and such that f (x) - g (x) = 7 (^ means exponent)
Suppose that derivative of the function f is given by f'(x) = 3x and suppose that the point (4,2) is on the graph of f. Write an equation of a line tangent to the graph of f at the point (4,2)
Hi, this was the question h is a function such that h(0) = 1, h(2) = 7, h(4) = 5, h'(0) = -2, h'(2) = 3, and h'(4) = -1 Evaluate lim h(w)−1 w→0 ¯¯¯w¯¯¯ and this was the response: lim h(w)−1 w→0 ¯¯¯w¯¯¯
Let n(x) = 1/x. Find a formula for n''(x)
Let g(x) = x^2 + x. Use the limit definition of the derivative to show that g'(x) = 2x + 1
Differentiate: Y=x^2+4x(x^3/2) NOTE: x^2 =(squared) x^3/2 =(x raised to the 3/2) Determine whether the limit exist, If so compute the limit: Lim (square root of x) minus 4 divided by x^3+18 x-->1 NOTE: x^3 = x cubed Find the Limit: Lim x--> 0 (1 divided by x^3 minus 1) +1 NOTE:
Q#8. For f(x)=2-5X^2, Find: i) f`(x). Answers: a. -5x b. -10x c. -10x d. 0 Q#9. Consider f(x)=3-4*Square root(X) i) Find f`(x) Answers: a. - square root of X / 2 b. - 4 / square root of X c. -2/square root of x d. -8 / Square root of x. Q#10. Consider f(x)=3x/x+9 i) find f`(x) Q#7. A per
Given the equation y = x , determine the equation for the derivative dy/dx, using x-5 the delta process or the definition. Determine the derivative y = ( 3x+4)^ (2/3) * (x-1), find dy/dx Given the implicit relation x^3 y^2 + 2y^3 = 4x^3 + 7, determine dy/dx A particle moves in metres according to the parameterized
27. A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10 m/s. It strikes the ground with a speed of 60 m/s. How tall is the building? Answer: y0 = 178.57 m 35. A stone is dropped from rest at an initial height h above the surface of the Earth. Show that the speed wit
Derivation using the product rule. See attached file for full problem description.
Early one morning it began to snow at a constant rate. At 7 AM a snowplow set off to clear a road. By 8 AM it had traveled 2 miles but it took two more hours for the snowplow to go another 2 miles. Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour), at what time did it start to snow?
1. The resale value of a certain industrial machine decreases at a rate that depends on its age. When the machine is t years old, the rate at which its value is changing is -960e^-t/5 dollars per year. a) Express the value of the machine in terms of its age and initial value. b) If the machine was originally worth $5,200,
^ Find the directional derivative of f at P in the direction of a , where f(x,y) = x^2 - 3xy + ^ 4y^3 ; P (-2,0) ; a = <1,2> .
Verify: If f(x, y, z) = 2z^(3) - 3 (x^(2) + y^(2)) z , then F_xx + F_yy + F_zz = 0 . P.S. F_xx = F(subscript)xx, etc.....
^ ^ ^ ^ Calculate d/dt [ r_1(t) * r_2(t) ] for r_1(t) = < 2t, 3t^(2), t^(3) > , r_2(t) = < 0, 0, t^(4)> . P.S. * = cross product and r_1 = r(subscript)1, r_2 = r(subscript)2 keywords: differentiating, differentiate, di
Show that z = x^(2) - y^(2) + 2xy satisfies Laplace's Equation z (subscript)xx + z (subscript)yy .
Verify: If w = tan (x^(2) + y^(2)) + x(y)^(1/2) , then w (subscript)xy = w (subscript)yx . keywords: differentiating, differentiate, differential
^ ^ ^ Find k(t) for r(t) = t^(2) i + t^(3) j . keywords: differentiating, differentiate, differential
^ ^ ^ Given r = 3 cos t i = 3 sin t j , find the velocity, acceleration and speed for an arbitrary time t. keywords: differentiating, differentiate, differential