University:
Bismarck State College
Course:
MATH 146 | Applied Calculus I
Academic year:

2023
Views:

270

Pages:

4
Author:

Alexa Livingston

1. If y ( x ^ 2 1 )^ 3 then d/dx (y) ? 3 * (x ^ 2 + 1) ^ 2 * 2x = 6x * (x ^ 2 + 1) ^ 2 Chain 2. If Quotient then? dx (12x + 9 - 12x - 16)/((4x + 3) ^ 2) = - 7/((4x + 3) ^ 2) y = (3x + 4)/(4x + 3) ((3x + 4)' * (4x + 3) - (4x + 3)' * (3x + 4))/((4x + 3) ^ 2) = (3(4x + 3) - 4(3x + 4))/((4x + 3) ^ 2) 3. a) lim x infty ( lambda x^ 2 -4x+5 3y * x ^ 2 + 9x - 2 = 2 1/3 b) lim x -> ∞ (x - 4)/(3x ^ 2 + 9x - 2) = c) DNE lim x -> ∞ (x ^ 3 - 4x + 5)/(3x ^ 2 + 9x - 2) = (x+1)-(x-1 d) lim-(1)봄 ロー = lim = lim 2 2x 2 4. Draw a graph if f(x) > 0 f' * (x) < 0 and f^ prime prime (x) > 0 . above X-axis Decreasing concave up 5. a) (cos x)' = - sin x lim h -> 0 (cos(x + h) - cos x)/h = b) (sinx)' =co x lim h -> 0 (sin(x + h) - sin x)/h = c) lim h -> 0 ((x + h) ^ 3 - x ^ 3)/h = (x ^ 3) ^ 1 = 3x ^ 2 { Definition of a Derivative 6. If x ^ 2 + y ^ 2 = 100 what is the value of at the point (-6, 8)? (d ^ 2 * y)/(d * x ^ 2) 2x + 2y * d/dx (y) = 0 2y * d/dx (y) = - 2x d/dx (y) = - x/y (d ^ 2 * y)/(d * x ^ z) = ((- x)' * y - y' * (- x))/(y ^ 2) = (- y + x(- x/y))/(y ^ 2) = - (y + x * (dy)/(dx))/(y ^ 2) = (- 8 - 36/8)/64 = (- 8 - 9/2)/64 = - 25/2 * 1/64 = boxed -25 128 d^ 2 y dx^ 2 | (- b, 9) = (- 8 + (- 6)(- 6/8))/(pi ^ 2) 7. acceleration attained on the interval [0, 3] by the particle whose velocity is given by v(t) = t ^ 3 - 3t ^ 2 + 12t + 4 Extreme Value Theorem a(t) = 3t ^ 2 - 6t + 12 Endpoints: a(0) = 12 a(3)=3( 37 - 6(3) + 12 27-18-12 = 21max Critical Points: o = 6t - 6 t = 1 9 Minimum a' * (t) = 6t - 6 a(1) = 3 * (1) ^ 2 - 6(1) + 12 = 3 - 6 + 12 3. Letfbe the function f(x) = 2x ^ 3 + 3x ^ 2 - 12x + 1 On what intervals is the function both increasing and concave down? f' * (x) = 6x ^ 2 + 6x - 12 f^ prime prime (x) = 12x + 6 = 6(2x + 1) = 6(x ^ 2 + x - 2) = 6(x + 2)(x - 1) X f' * (x) 2 F^ prime prime (x) + → - 1/2 x + 2 x < - 2 9. The derivative g' of a function g is continuous and has exactly two zeros. Selected values of g' are given in the table below. If the domain of g is the set of all real numbers, then g is increasing on which intervals? g' * (x) -4 -3 -2 - -3 0 2 -101234 3210-3-2 g' * (x) > 0 - 2 < x < 2 10. f(x) = 2x + 1 x <= 2; 4x - 3 x > 2 a) Is f continuous at x = 2' Justify your answer. b) Is f differentiable at x = 2% Justify your answer. 2(2)+1=5 4(2)-35 yes continuous lim x -> z f(x) = lim x -> 2 f(x) no, left and right hand derivatives are not the same f^ prime (x)=\ [[2, x < 2], [4, x > 2]] 11. Given the graph of f^ prime prime (x) below: f^ prime prime (x) -3-2-1 1 234567 X4 For what values of x does the graph of fhave a point of inflection? x = - 3 , x = 1 , x = 4 lceil^ prime prime (x) switches signs 12. If g(x) = x ^ 2 * f(x) find an equation of the tangent line to the graph of g(x) if x = 2 f(2) = 7 and f' * (2) = - 3 Pant: g(z) = z ^ 2 * f(z) slope: g' * (x) = (x ^ 2)' * f(x) + f' * (x) * x ^ 2 28 g'(x) = 2x f(x) + F'(x) x² g' * (z) = z(z) * f(z) + f' * (z) * (z) ^ 2 = 4(7) + (- 3)(4) = 28-12 = 16 y - 28 = 16(x - 2) 13. 2 778 Graph of f -2 The graph of the function f in the figure above consists of four line segments. Let g be the function defined by g(x) = integrate f(t) dt from 0 to x Write an equation of the line tangent to the graph of g at x = 4 . g(4) = integrate f(t) dt from 0 to 4 = - 1/2 g' * (x) = f(x) g' * (4) = f(4) = - 2 y + 1/2 = - 2(x - 4)