Calculus Solutions

Gegeben ist die Funktion f mit f(x) = 1/10 * (1/4 * x^4 - 9/2 * x^2) und der Definitionsmenge D_f = [-3; 4]. 1.1 Untersuchen Sie den Graphen G_f auf Symmetrie. 1.2 Bestimmen Sie die Schnittpunkte von G_f mit den Koordinatenachsen. 1.3 Ermitteln Sie das Monotonieverhalten der Funktion f. 1.4 Berechnen Sie die Koordinaten und die genaue Art aller Extrempunkte von G_f. Geben Sie auch die Wertemenge W_f an. 1.5 Zeichnen Sie den Graphen unter Verwendung aller bisherigen Ergebnisse in ein kartesisches Koordinatensystem.

Answer: 1.1 f is even → graph is symmetric about the y-axis (within

Oct 10, 2026

(9 points each part) Find the derivative of the function. PLEASE do not simplify. Label. a) f(x) = 1/(2x-4)^3 b) f(x) = ((2x+3)/(5x-2))^3 c) f(x) = (6x-7)^2 * (6x^2+2)^3 d) f(x) = sin(2x^3 + 2x) e) f(x) = (x^3 + 2x + 1)/(2x+1)^2 f) f(x) = (2x)cos^2(3x) g) f(x) = cos(tan(x)) h) If y = sin 2x, then dy/dx =

Answer: a) $f'(x) = -3(2x-4)^{-4} \cdot (2)$ b) $f'(x) = 3\left(\fra

Sep 28, 2026

Suppose that f(2) = -1, g(2) = 5, f'(2) = -4, and g'(2) = 3, find the following: a) (f - g)'(2) b) (fg)'(2) c) (f/g)'(2) d) (f/(f + g))'(2)

Answer: a) $(f-g)'(2) = -7$ b) $(fg)'(2) = -23$ c) $(f/g)'(2)

Sep 21, 2026

Find the derivative of the function. PLEASE do not simplify. Label. a) f(x) = (1/2)*pi*x^5 - 1/x^6 b) f(x) = x / (4 - x^3) c) f(x) = (4x^2 - 3x + 1)(-2x + 7) d) f(x) = cot(x) / sin(x) e) f(x) = -3cos(x) + 4sin(x) f) f(x) = tan(x) + cos(x) g) f(x) = x^6 * sec(x)

Answer: a) $f'(x) = \frac{5}{2}\pi x^4 + 6x^{-7}$ | b) $f'(x) = \fra

Sep 21, 2026

Find the derivative of the function. PLEASE do not simplify. Label. a) f(x) = (1/2)*pi*x^5 - 1/x^6 b) f(x) = x/(4 - x^3) c) f(x) = (4x^2 - 3x + 1)(-2x + 7) d) f(x) = cot(x)/sin(x) e) f(x) = -3cos(x) + 4sin(x) f) f(x) = tan(x) + cos(x) g) f(x) = x^6 * sec(x)

Answer: a) $f'(x) = \frac{1}{2}\pi \cdot 5x^{4} - (-6)x^{-7}$ | b) $

Sep 21, 2026

The image shows a velocity-time graph where: - The y-axis (velocity) ranges from -6 to 5 - The x-axis (time) ranges from 0 to 11 - Initial velocity is known - Initial position is known From the graph, the velocity function appears to be: - v = 5 from t = 0 to t = 1 - v decreases linearly from 5 to -6 from t = 1 to t = 5 - v = -6 from t = 5 to t = 9 - v increases from -6 to 0 from t = 9 to t = 11 Given a velocity-time graph with initial velocity and initial position known, find the position function x(t). Find the displacement starting from 5 seconds to 11 seconds

Answer: The displacement from t = 5 to t = 11 is $\Delta x = -30$ un

Sep 21, 2026

The image shows a velocity-time graph where: - The y-axis (velocity) ranges from -6 to 5 - The x-axis (time) ranges from 0 to 11 - Initial velocity is known - Initial position is known From the graph, the velocity function appears to be: - v = 5 from t = 0 to t = 1 - v decreases linearly from 5 to -6 from t = 1 to t = 5 - v = -6 from t = 5 to t = 9 - v increases from -6 to 0 from t = 9 to t = 11 Given a velocity-time graph with initial velocity and initial position known, find the position function x(t).

Answer: $x(t) = \begin{cases} x_0 + 5t & 0 \le t \le 1 \\ x_0 + 5 +

Sep 21, 2026

dy/dx = (x + y)/x, using the homogeneous substitution v = y/x, what is y'?

Answer: After substitution, $y' = \frac{dy}{dx} = v + x\frac{dv}{dx}

Sep 21, 2026

(2x + y) dx + (x + 2y) dy = 0

Answer: $x^2 + xy + y^2 = C$, where C is an arbitrary constant.

Sep 21, 2026

dy/dx + 2y = x

Answer: $y = \dfrac{x}{2} - \dfrac{1}{4} + Ce^{-2x}$, where C is an

Sep 21, 2026

Find the x-coordinate(s) on the curve y = x - x^2 where the tangent line is horizontal.

Answer: $x = \dfrac{1}{2}$

Sep 14, 2026

Differentiate (find the derivative of) the function. Be sure to label correctly. DO NOT SIMPLIFY ANSWERS. (7 points each) a) f(x) = sqrt(x) - 1/((x^2)^(1/3)) (simplify first on the second term - do not use quotient rule) b) f(x) = x*sqrt(x) (simplify first - do not use product rule) c) f(x) = (x^3 - x^2 + 4)/x (simplify first - do not use quotient rule) d) f(x) = (7x^2 + x + 1)(3x + 2) (Use product rule) e) f(x) = (e^x + 8x)(4x^(-8) - x) (use product rule)

Answer: a) $f'(x) = \frac{1}{2}x^{-1/2} + \frac{2}{3}x^{-5/3}$ | b)

Sep 14, 2026

3. Find the value of the following limits if they exist. Graph with calculator to help with a - d. (These four are 4 pts each.) a) lim_{x->0^+} sqrt(x) = b) lim_{x->-4} sqrt(16 - x^2) = c) lim_{x->4^-} 1/(x - 4) = d) lim_{x->3^-} |x - 3|/(x - 3) = SHOW ALL WORK TO JUSTIFY your answers to e - g. (These three are 5 pts each) e) lim_{x->1} (x^2 - x)/(2x^2 + 5x - 7) (use factoring)

Answer: a) $\lim_{x \to 0^+} \sqrt{x} = 0$ | b) $\lim_{x \to -4} \sq

Sep 8, 2026

Find the derivative of f(x) = x³ - 4x² + 7

Answer: f'(x) = 3x² - 8x

Apr 12, 2026